Question:
What is m $\angle $ I?

m $\angle $ I= $\Box$ °

Answer: m \$\angle \$ I=36°

1. Let the figure be circle O. According to the definition of diameter, IK is the diameter of the circle. According to the definition of inscribed angle, ∠I, ∠J, and ∠K are inscribed angles. According to the corollary of the inscribed angle theorem (corollary 2), the inscribed angle subtended by the diameter is a right angle, so ∠J is a right angle, i.e., ∠J = 90°.

2. According to the triangle angle sum theorem, the sum of the three angles in a triangle is 180°, we get ∠I + ∠J + ∠K = 180°.

4. Knowing ∠J = 90° and ∠K = 54°, substituting these into the above equation, we get ∠I + 54° + 90° = 180°.

5. Simplifying the above equation, we get ∠I + 144° = 180°.

6. After calculation, ∠I = 180° - 144° = 36°.

7. Through the above reasoning, the final answer is 36°.

Definition of Diameter

<note>KJ is the diameter</note>, connecting the center of the circle and the points K and J on the circumference.

Definition of Inscribed Angle

The vertex I of angle IJK lies on the circumference of the circle, and the two sides of angle IJK intersect the circle at points J and K respectively. Therefore, <note>angle IJK is an inscribed angle</note>.

Triangle Angle Sum Theorem

<note>In triangle IJK, angles I, J, and K are the three interior angles of triangle IJK</note>, according to the Triangle Angle Sum Theorem, <note>∠I + ∠J + ∠K = 180°</note>.

<note>In the figure of this problem</note>, in circle O, <note>the inscribed angle ∠J subtended by the diameter IK is a right angle (90 degrees)</note>, that is, <note>∠J = 90°</note>.

Definition of Right Triangle

A triangle is classified as a right triangle if it has one interior angle measuring exactly 90 degrees.

In the figure of this problem, in triangle IJK, <note>angle IJK is a right angle (90 degrees)</note>, therefore <note>triangle IJK is a right triangle</note>. <note>Side IJ and side JK are the legs</note>, <note>side IK is the hypotenuse</note>.


Question:
The radius of a circle is 1 inch. What is the length of a 45° arc?

Give the exact answer in simplest form.
 $\Box$ 𝜋/ inches

Answer: 1/4𝜋 inches

1. Given the radius r of the circle is 1 inch, and the angle is 45°.

2. According to the formula for the circumference of a circle C = 2πr, and substituting r = 1, we get C = 2π(1) = 2π.

3. Determine the formula for the arc length as l = C * (m/360), where C is the circumference and m is the degree measure of the arc.

4. Given that the degree measure of the arc m is 45°, substitute the known values into the arc length formula to get l = 2π * (45/360).

5. Simplify the expression, 45/360 = 1/8, thus l = 2π * (1/8).

6. Calculate to get l = π/4.

7. Through the above reasoning, the final answer is π/4.

Definition of Circle

A circle is the locus of all points in a plane that are at a fixed distance from a given point.

<note>Point O is the center of the circle</note>, <note>the radius is 1 inch</note>. All points in the figure that are <note>1 inch away from point O</note> are on the circle.

Circumference Formula of Circle

The circumference of a circle is the length around the circular boundary. It can be calculated using the formula \( C = 2\pi r \), where \( C \) represents the circumference and \( r \) represents the radius.

<note>Point is the center of the circle</note>, <note>line segment is the radius r</note>. According to the circumference formula of the circle, <note>the circumference C of the circle is equal to 2π multiplied by the radius r</note>, that is, C=2πr, where <note>r=1 inch</note>, therefore C=2π(1)=2π inches.

Definition of Arc

An arc is a curve on a circle that is defined by two distinct points on the circle.

In the figure of this problem, there are <note>two points forming a 45° angle with the center of the circle</note> on the circle, and this arc is <note>a segment of a curve connecting these two points</note>. According to the <note>definition of arc</note>, this arc is <note>a segment of a curve between two points on the circle</note>.

Arc Length Formula of a Circle

The arc length refers to the length of a segment of the circumference of a circle. It can be calculated using the formula 𝓁 = C × (m/360), where m is the measure of the central angle in degrees.

<note>In a circle, the degree measure of the central angle of the arc is 45 degrees</note>, <note>the circumference is 2π</note>. According to the <note>Arc Length Formula of a Circle 𝓁 = C × (m/360)</note>, the arc length can be expressed as <note>𝓁 = C × (m/360)</note>. Therefore, the arc length is equal to <note>the circumference 2π multiplied by the degree measure of the central angle 45°</note> divided by <note>360 degrees</note>.


Question:
△QRS and △UVW are shown below.

Which statement is true?

- △QRS is similar to △UVW.
- △QRS is not similar to △UVW.
- There is not enough information to determine whether the triangles are similar.

Answer: - △QRS is not similar to △UVW.

1. Given △QRS, ∠Q = 104°, ∠R = 23°.

2. According to the triangle angle sum theorem, the sum of the interior angles of a triangle is 180°. Therefore, in △QRS, ∠Q + ∠R + ∠S = 180°.

3. Substituting the given angles into the equation, we get 104° + 23° + ∠S = 180°.

4. Solving the equation, we get ∠S = 180° - 127° = 53°.

5. Therefore, the three interior angles of △QRS are ∠Q = 104°, ∠R = 23°, ∠S = 53°.

6. In the given data, for △UVW, ∠U = 49°, ∠V = 23°.

7. If △QRS is similar to △UVW, then they should have two corresponding angles that are equal.

8. By checking whether there are two equal angles, it can be observed that the largest angle in △UVW is 49° and the closest angle to 49° in △QRS is 53°, which are not equal.

9. Therefore, at least one necessary condition is not satisfied, which means △QRS and △UVW cannot be similar.

10. Based on the above reasoning, the final answer is △QRS is not similar to △UVW.

Definition of Triangle

A geometric figure is called a triangle if it is formed by three non-collinear points and the line segments connecting them.

<note>A triangle QRS is a geometric figure composed of three non-collinear points Q, R, S and their connecting line segments QR, RS, SQ</note>. Points <note>Q, R, S are the three vertices of the triangle</note>, and line segments <note>QR, RS, SQ are the three sides of the triangle</note>. Similarly, <note>a triangle UVW is a geometric figure composed of three non-collinear points U, V, W and their connecting line segments UV, VW, WU</note>. Points <note>U, V, W are the three vertices of the triangle</note>, and line segments <note>UV, VW, WU are the three sides of the triangle</note>.

Similarity Theorem for Triangles (AA)

Two triangles are similar if two of their corresponding angles are congruent.

In <note>triangle QRS</note> and <note>triangle UVW</note>, if <note>angle Q is equal to angle U</note> and <note>angle R is equal to angle V</note>, then <note>triangle QRS is similar to triangle UVW</note>. Based on the known angles (<note>∠Q = 104°</note>, <note>∠R = 23°</note>, <note>∠S = 53°</note> and <note>∠W = 49°</note>, <note>∠V = 23°</note>), it can be determined that the conditions of this theorem are not met.

Triangle Angle Sum Theorem

The sum of the interior angles of any triangle is 180°.

In the diagram of this problem, in triangle QRS, <note>angle Q</note>, <note>angle R</note>, and <note>angle S</note> are the three interior angles of triangle QRS. According to the Triangle Angle Sum Theorem, <note>angle Q + angle R + angle S = 104° + 23° + angle S</note> = 180°. The same logic applies to triangle UVW, <note>angle U</note>, <note>angle V</note>, and <note>angle W</note> are the three interior angles of triangle UVW, <note>angle U + angle V + angle W = 49° + 23° + angle U</note> = 180°.


Question:
The radius of a circle is 3 meters. What is the length of a 45° arc?

Give the exact answer in simplest form.
 $\Box$ 𝜋/ meters

Answer: 3/4𝜋 meters

1. Given that the radius of the circle is 3 meters, we first need to calculate the circumference of this circle.

2. According to the formula for the circumference of a circle C = 2πr, where r is the radius of the circle, we substitute r = 3 to get C = 2π * 3.

3. Calculating this, we get the circumference C = 6π meters.

4. The problem requires calculating the length of a 45° arc.

5. According to the formula for the length of an arc l = C * (θ/360), where θ is the central angle in degrees, we substitute C = 6π and θ = 45.

6. Calculating the arc length l = 6π * (45/360).

7. Simplifying the fraction, we get 45/360 = 1/8.

8. Therefore, the arc length l = 6π * 1/8.

10. Through the above reasoning, we finally obtain the answer as 3π/4 meters.

Definition of Radius

A radius is a line segment joining the center of a circle to any point on the circumference of the circle.

<note>The radius of the circle is 3 meters, denoted as r</note>. <note>The center of the circle is the orange point in the diagram</note>, <note>The distance from any blue point on the circumference to the orange point is 3 meters</note>. <note>The length of the line segment from the orange point to any blue point on the circumference is 3 meters</note>, therefore <note>this line segment is the radius of the circle</note>.

Definition of Central Angle

An angle formed by two radii connecting two points on the circumference of a circle to the center of the circle is called a central angle.

In the figure of this problem, <note>the center of the circle is point O</note>, <note>the orange line segment is the line connecting two points on the circle from the center O</note>. <note>The angle ∠AOB between the orange line segments is 45°</note>, which is called the central angle.

Definition of Arc

In the figure of this problem, there are two points on the circle which are <note>the two endpoints of the orange segment</note>, <note>the arc is a curve connecting these two points</note>. According to the definition of the arc, this arc is a segment of the curve between two points on the circle, and <note>the corresponding central angle is 45°</note>.

Circumference Formula of Circle

<note>The center of the circle is point O</note>, the segments <note>OA and OB are the radius r</note>. According to the <note>Circumference Formula of Circle</note>, the circumference of the circle <note>C is equal to 2π multiplied by the radius r</note>, that is, <note>C=2πr</note>, where <note>r=3 meters</note>, therefore <note>C=2π*3=6π meters</note>.

Arc Length Formula of a Circle

In the figure of this problem, <note>the central angle θ is 45°</note>, <note>the circumference C is 6π meters</note>, therefore <note>the arc length l = 6π * (45/360) = 3π/4 meters</note>.


Question:
What is m $\angle $ D?

m $\angle $ D= $\Box$ °

Answer: m \$\angle \$ D=58°

1. Given that DE is the diameter of the circle, according to the definition of the inscribed angle and (the corollary 2 of the inscribed angle theorem) the inscribed angle subtended by the diameter is a right angle, it can be concluded that ∠DFE is a right angle. (The corollary 2 of the inscribed angle theorem) the inscribed angle subtended by the diameter is a right angle states that the inscribed angle subtended by the diameter is a right angle.

2. According to the definition of a right triangle, triangle DEF is a right triangle, where ∠DFE is 90°.

3. In a triangle, the sum of the three interior angles is 180°. For triangle DEF, since ∠DFE = 90°, the sum of the remaining angles ∠EDF and ∠DEF should be 90°, that is, ∠EDF + ∠DEF = 90°.

4. Given ∠DEF = 32°, according to the above conclusion, we get 32° + ∠EDF = 90°.

5. Solving the equation 32° + ∠EDF = 90°, we can subtract 32° from both sides to get ∠EDF = 58°.

6. Through the above reasoning, the final answer is 58°.

Definition of Diameter

A diameter is a line segment that passes through the center of a circle and has its endpoints on the circle. It is the longest chord of the circle, with a length equal to twice the radius of the circle.

<note>DE is the diameter</note>, connecting the center of the circle and the points <note>D and E on the circumference</note>, with a length of <note>2 times the radius</note>, i.e., <note>DE = 2r</note>.

Definition of Inscribed Angle

An inscribed angle is an angle whose vertex lies on the circumference of a circle and whose sides intersect the circle at two points.

The vertex F of <note>angle DFE is on the circumference</note>, and <note>the two sides of angle DFE intersect the circle at points D and E</note> respectively. Therefore, <note>angle DFE is an inscribed angle</note>.

Corollary to the Inscribed Angle Theorem 2: The Angle Subtended by the Diameter

The angle subtended by the diameter of a semicircle is a right angle; conversely, if an inscribed angle is 90°, then the chord subtending that angle is a diameter.

The angle subtended by <note>diameter DE</note> is a <note>right angle (90 degrees)</note>.

Triangle Angle Sum Theorem

In the figure of this problem, <note>in triangle DEF, ∠DEF, ∠EDF, and ∠DFE are the three interior angles of triangle DEF</note>, according to the Triangle Angle Sum Theorem, <note>∠DEF + ∠EDF + ∠DFE = 180°</note>.


Question:
What is the measure of $\overset{\frown}{RS}$ ?

m $\overset{\frown}{RS}$ = $\Box$ °

Answer: m \$\overset{\frown}{RS}\$ =67°

1. Given that line segments QT and RS are chords on the corresponding circle, and both chords have a length of 48.

2. According to the central angle theorem, the central angles subtended by chords QT and RS are equal.

3. Based on the properties of central angles, the central angle subtended by chord QT and arc QT is 67°.

4. Since the lengths of chords QT and RS are equal, according to the central angle theorem, arcs QT and RS are equal.

5. According to the central angle theorem, the degree measure of the central angle subtended by arc RS is equal to the central angle subtended by arc QT, which is 67°.

6. Through the above reasoning, the final answer is 67°.

Definition of Circle

<note>In circle O, point O is the center of the circle</note>, <note>the radius is a fixed length</note>. All points in the figure that are at a distance equal to the fixed length from <note>point O are on circle O</note>.

Definition of Chord

A chord is defined as a line segment that connects any two points on a circle.

In the figure of this problem, in the circle, <note>points Q and T are any two points on the circle, and the segment QT connects these two points, so the segment QT is a chord of the circle</note>. Similarly, <note>points R and S are any two points on the circle, and the segment RS connects these two points, so the segment RS is a chord of the circle</note>.

Definition of Central Angle

In the diagram of this problem, in the circle, <note>point Q and point T are two points on the circle, the center of the circle is point O</note>. <note>The angle ∠QOT formed by the lines OQ and OT is called the central angle</note>. Similarly, <note>point R and point S are two points on the circle</note>, <note>the angle ∠ROS formed by the lines OR and OS is called the central angle</note>.

Property of Central Angle

The degree measure of an arc is equal to the degree measure of the central angle that subtends the arc.

The arc <note>QT</note> corresponds to the central angle <note>QOT</note>, and the degree measure of the arc <note>QT</note> is equal to the degree measure of the angle <note>QOT</note>.

Central Angle Theorem

In the same circle or in congruent circles, arcs corresponding to equal central angles are equal, chords corresponding to equal central angles are equal, and the perpendicular distances from the center to these chords are also equal.

<note>The chords QT and RS within the same circle are equal</note>, then <note>the central angles QOT and ROS are equal</note>, <note>the arcs QT and RS are equal</note>.

<note>The arcs QT and RS within the same circle are equal</note>, then <note>the central angles QOT and ROS are equal</note>, <note>the chords QT and RS are equal</note>.


Question:
The radius of a circle is 6 kilometers. What is the length of a 45° arc?

Give the exact answer in simplest form.
 $\Box$ 𝜋/ kilometers

Answer: 3/2𝜋 kilometers

1. Given the radius r = 6 km, first calculate the circumference of the entire circle using the formula C = 2πr.

2. Substitute the given radius value into the circumference formula, C = 2π × 6 = 12π km. Therefore, the circumference of the circle is 12π km.

3. The central angle corresponding to the arc length is 45°.

4. Use the arc length formula l = C × (m/360), where m is the degree measure of the central angle, and substitute the given values into the formula.

5. Substitute C = 12π and m = 45 into the arc length formula to get l = 12π × (45/360).

6. Further simplify the arc length calculation, l = 12π × 1/8 = 3π/2 km.

7. Through the above reasoning, the final answer is 3π/2 km.

Definition of Circle

<note>The point is the center of the circle</note>, <note>the radius is 6 kilometers</note>. All points in the diagram that are <note>6 kilometers away from the point are on the circle</note>.

Definition of Arc

<note>There are two points on the circle that are the endpoints of the arc</note>, <note>the arc is a segment of an orange curve connecting these two points</note>. According to the definition of an arc, <note>an arc is a segment of a curve between two points on a circle</note>.

Definition of Central Angle

<note>The center of the circle is point O</note>, the angle ∠AOB formed by the lines OA and OB from the points <note>A and B on the circle to the center</note> is called the central angle, and <note>this central angle is 45°</note>.

Circumference Formula of Circle

<note>The radius of the circle r = 6 kilometers</note>. According to the circumference formula of the circle, the circumference C is equal to 2π multiplied by <note>radius r</note>, that is, <note>C = 2πr</note>.

Arc Length Formula of a Circle

Original text: <note>Arc length l = 12π × (45/360)</note>. By calculation, the result is <note>l = 12π × 1/8 = 3π/2 kilometers</note>.


Question:
What is m $\angle $ G?

m $\angle $ G= $\Box$ °

Answer: m \$\angle \$ G=56°

1. Given that segment GH is the diameter of the circle, according to (Corollary 2 of the Inscribed Angle Theorem), the inscribed angle subtended by the diameter is a right angle, ∠GIH is a right angle (90°).

2. In ΔGHI, since ∠GIH is a right angle and equals 90°, according to the Triangle Sum Theorem, the sum of the three angles in the triangle is 180°.

3. Since ΔGHI is a right triangle and ∠GIH is 90°, it can be concluded that the sum of ∠GHI and ∠HGI is 90°.

4. Given that ∠GHI equals 34°, according to the conclusion in step 3, ∠HGI = 90° - ∠GHI.

5. Substituting the given condition, calculate ∠HGI = 90° - 34° = 56°.

6. Through the above reasoning, the final answer is ∠HGI = 56°.

Definition of Right Triangle

<note>In the diagram of this problem, in triangle GHI, angle GIH is a right angle (90 degrees)</note>, therefore triangle GHI is a right triangle. <note>Side GI and side HI are the legs</note>, <note>side HG is the hypotenuse</note>.

Triangle Angle Sum Theorem

<note>In the figure of this problem, in triangle GHI, angle GIH, angle GHI, and angle HGI are the three interior angles of triangle GHI</note>, according to the <note>Triangle Angle Sum Theorem</note>, <note>angle GIH + angle GHI + angle HGI = 180°</note>.

Definition of Inscribed Angle

<note>The vertex I of angle GIH is on the circumference</note>, <note>the two sides of angle GIH intersect the circle at points G and H respectively</note>. Therefore, angle GIH is an inscribed angle.

<note>In a circle, the inscribed angle HIG subtended by the diameter HG is a right angle (90 degrees)</note>.


Question:
What is the length of $\overline{UV}$ ?

UV= $\Box$

Answer: UV=14

1. Given: There are chords \\overline{TW} and \\overline{UV} on the circumference, and the corresponding arcs are congruent, i.e., arc \\overset{\\frown}{WT} \\cong \\overset{\\frown}{UV}.

2. According to the central angle theorem, if two arcs are equal, then the chords corresponding to them are also equal.

3. Therefore, \\overline{TW} \\cong \\overline{UV}, which means the length of \\overline{TW} is equal to the length of \\overline{UV}.

4. Given \\overline{TW} = 14, the length of \\overline{UV} is also 14.

5. Through the above reasoning, the final answer is: the length of UV is 14.

Definition of Arc

<note>In the figure of this problem, there are two points W and T on the circumference, arc \overset{\frown}{WT}</note> is a segment of the curve connecting these two points; <note>there are two points U and V on the circumference, arc \overset{\frown}{UV}</note> is a segment of the curve connecting these two points. According to the definition of arc, <note>arc \overset{\frown}{WT}</note> is a segment of the curve between two points W and T on the circle, <note>arc \overset{\frown}{UV}</note> is a segment of the curve between two points U and V on the circle.

Definition of Chord

In the figure of this problem, in the circle, <note>points T and W are any two points on the circle</note>, <note>the line segment \overline{TW} connects these two points</note>, so <note>the line segment \overline{TW} is a chord of the circle</note>. Similarly, <note>points U and V are any two points on the circle</note>, <note>the line segment \overline{UV} connects these two points</note>, so <note>the line segment \overline{UV} is also a chord of the circle</note>.

Central Angle Theorem

<note>Arcs WT and UV in the same circle are equal</note>, <note>the corresponding chords WT and UV are equal</note>.


Question: As shown in the figure, the height of the rectangular prism is 10cm. Calculate the volume of the rectangular prism in the figure.

Answer: \$1690cm^2\$

1. Given that the height of this rectangular prism is 10 cm, and the base is a square (according to the definition of a rectangle, the four sides of the base of the rectangular prism are equal, which meets the definition of a square), denoted as ABCD, with each side length being 13 cm.

2. According to the volume formula of the prism: Volume = Base area × Height, where the base is a square, the base area is obtained using the square area formula as 13 cm × 13 cm.

3. Calculate the base area: 13 × 13 = 169 square centimeters.

4. Based on steps 2 and 3, the volume is: 169 square centimeters × 10 centimeters = 1690 cubic centimeters.

5. Through the above reasoning, the final answer is 1690 cubic centimeters.

Definition of Square

A quadrilateral is a square if and only if all four sides are congruent (≅) and all four interior angles are right angles (90°).

<note>Square ABCD</note> has <note>sides AB, BC, CD, and DA equal</note>, and <note>angles A, B, C, and D are all right angles (90 degrees)</note>. Therefore, <note>Square ABCD</note> has <note>four equal sides and four right angles (90 degrees)</note>, meeting the definition of a square.

Area Formula for Square

The area of a square is equal to the square of its side length, that is, Area = side length × side length.

In the square ABCD, side AB is one of the sides of the square, with a side length of 13. Therefore, according to the area formula for a square, <note>the area of square ABCD A = 13²</note>.

Volume Formula of Prism

The volume of a prism is equal to the base area multiplied by the height.

The area of the base ABCD is <note>169</note>, the height of the prism is <note>10</note>. Therefore, according to the volume formula of the prism, the volume of the prism is equal to <note>base area 169 multiplied by height 10</note>.

Definition of Rectangle

A quadrilateral is called a rectangle if and only if each of its interior angles is a right angle (90°) and its opposite sides are both parallel (∥) and congruent (≅).

<note>The base is a square</note>. The side length of the base is <note>13cm</note>, and each angle is a <note>right angle (90 degrees)</note>, with opposite sides parallel and equal in length.


Question:
The radius of a circle is 3 centimeters. What is the length of a 90° arc?

Give the exact answer in simplest form.
 $\Box$ 𝜋/ centimeters

Answer: 3/2𝜋 centimeters

1. Given the radius of the circle is 3 cm. According to the formula for the circumference of a circle C=2πr, where r is the radius of the circle, we get C=2π×3=6π cm.

2. The circumference of the circle is 6π cm. Since the circumference represents the length of the entire circle, the 360° of the circle corresponds to the circumference of 6π cm.

3. The problem requires calculating the length of the 90° arc. According to the arc length formula, the relationship between the arc length l, the circumference C, and the central angle m is l = C × (m/360).

4. Substitute the given conditions into the arc length formula: given C=6π cm and m=90°, therefore the arc length l = 6π × (90/360).

5. Simplify the calculation of the arc length: l = 6π × (1/4) = (6π/4) = (3π/2).

6. Through the above reasoning, the final answer is 3π/2.

Definition of Circle

A circle is the locus of all points in a plane that are at a fixed distance from a given point.

<note>The center of the circle is at the center of the diagram</note>, marked as a point, <note>the radius is 3 cm</note>. All points in the diagram that are 3 cm away from the center are on this circle.

Definition of Radius

In the figure of this problem, <note>the radius of the circle is marked as r=3 cm</note>, indicating that <note>the length of the line segment from the center of the circle to any point on the circumference is 3 cm</note>.

Definition of Arc

An arc is a curve on a circle that is defined by two distinct points on the circle.

<note>The purple part represents a 90° arc</note>, two points on the circumference connected by a 90-degree angle forming the arc between these two points. According to <note>the definition of arc</note>, the purple arc is a segment of the curve between two points on the circle.

Circumference Formula of Circle

The circumference of a circle is the length around the circular boundary. It can be calculated using the formula \( C = 2\pi r \), where \( C \) represents the circumference and \( r \) represents the radius.

Original: <note>The radius r of the circle is 3 cm</note>, the circumference C of the circle is equal to <note>2π multiplied by the radius r</note>, which is <note>C=2πr</note>. By calculating <note>C = 2π × 3 = 6π cm</note>, the complete circumference of the circle can be obtained as <note>6π cm</note>, representing the <note>green line circumference length</note> in the diagram.

Arc Length Formula of a Circle

The arc length refers to the length of a segment of the circumference of a circle. It can be calculated using the formula 𝓁 = C × (m/360), where m is the measure of the central angle in degrees.

<note>In this problem, by calculating the 90° arc length l = 6π × (90/360)=3π/2 cm</note>, the final arc length of 3π/2 cm can be obtained, <note>indicating the length of the purple arc in the figure</note>.


Question:
What is the measure of $\overset{\frown}{TW}$ ?

m $\overset{\frown}{TW}$ = $\Box$ °

Answer: m \$\overset{\frown}{TW}\$ =89°

1. Observe the figure, it is known that the two chords UV and TW inside the circle are equal, i.e., chord UV ≅ chord TW.

2. According to the central angle theorem: In the same circle or in equal circles, if two chords are equal, then their corresponding central angles are equal, hence their corresponding arcs are also equal.

3. From this property, we conclude that arc TW ≅ arc UV.

4. Given that the degree measure of arc UV is 89°, according to the conclusion in step 3, the degree measure of arc TW is the same as that of arc UV.

5. Through the above reasoning, we finally conclude that the degree measure of arc TW is 89°.

Definition of Arc

In the figure of this problem, there are two points <note>T and W</note> on the circle, and <note>arc TW</note> is a segment of the curve connecting these two points. There are two points <note>U and V</note> on the circle, and <note>arc UV</note> is a segment of the curve connecting these two points. According to the definition of an arc, <note>arc TW</note> is a segment of the curve between the two points <note>T and W</note> on the circle, and <note>arc UV</note> is a segment of the curve between the two points <note>U and V</note> on the circle.

Definition of Chord

<note>Point U and point V are any two points on the circle</note>, the line segment <note>UV</note> connects these two points, so the line segment <note>UV</note> is a chord of the circle; similarly, <note>point T and point W are any two points on the circle</note>, the line segment <note>TW</note> connects these two points, so the line segment <note>TW</note> is a chord of the circle.

Central Angle Theorem

<note>In the diagram of this problem, the chords UV and TW within the same circle are equal</note>, then <note>the central angles UOV and WOT are equal</note>, <note>the arcs UV and TW are equal</note>.


Question:
The radius of a circle is 7 kilometers. What is the length of a 180° arc?

Give the exact answer in simplest form.
 $\Box$ 𝜋/ kilometers

Answer: 7𝜋 kilometers

1. Given the radius r = 7 km, according to the circumference formula C = 2πr, we can calculate the circumference of the circle.

2. Substituting the radius into the circumference formula, we get: C = 2π(7) = 14π km.

3. Given that the central angle of the arc to be determined is 180°. According to the arc length formula l = C * (m/360), where m is the degree measure of the central angle.

4. Substituting the known values, we get the arc length l = 14π * (180/360).

5. Simplifying the fraction, 180/360 = 1/2.

6. Therefore, l = 14π * (1/2) = 7π.

7. Through the above reasoning, the final answer is 7π km.

Definition of Circle

<note>The center of the circle is the blue point in the diagram</note>, and the radius is 7 kilometers. All points in the diagram that are 7 kilometers away from <note>this blue point</note> are on the circle.

Definition of Radius

In the figure of this problem, in the circle, <note>point O is the center of the circle</note>, <note>point A and point B are any points on the circle</note>, <note>line segment OA and line segment OB are line segments from the center of the circle to any point on the circle</note>, therefore <note>line segment OA and line segment OB are the radii of the circle</note>, and the length is 7 kilometers.

Circumference Formula of Circle

<note>The radius of the circle r = 7 kilometers</note>. According to the circumference formula of the circle, the circumference of the circle <note>C is equal to 2π multiplied by the radius r</note>, which means <note>C = 2π * 7 = 14π kilometers</note>.

Definition of Central Angle

<note>In the diagram of this problem, the angle formed by the lines connecting two points on the circle to the center of the circle is called the central angle</note>. The degree of the <note>central angle in the diagram is 180°</note>.

Arc Length Formula of a Circle

In the figure of this problem, <note>the central angle is known to be 180°</note>, <note>the circumference C = 14π kilometers</note>, substituting into the formula gives: <note>l = 14π * (180/360) = 7π kilometers</note>. Therefore, the arc length is 7π kilometers.


Question:
The radius of a circle is 3 inches. What is the length of a 45° arc?

Give the exact answer in simplest form.
 $\Box$ 𝜋/ inches

Answer: 3/4𝜋 inches

1. The problem gives the radius of the circle as 3 inches, and we need to calculate the circumference of the circle.

2. According to the circumference formula: C = 2𝜋r, in this problem r = 3, therefore C = 2 × 𝜋 × 3 = 6𝜋 inches.

3. It is known that the arc length formula is: 𝓁 = C × m/360, where 𝓁 is the arc length and m is the central angle in degrees.

4. The central angle of the arc in the problem is 45°, so we substitute m = 45 into the arc length formula.

6. Simplify the value of 𝓁: 6𝜋 × 45/360 = 6𝜋 × 1/8 = 3𝜋/4 inches.

7. Through the above reasoning, we finally obtain the answer as 3𝜋/4 inches.

Definition of Circle

In the figure of this problem, in the definition of a circle, <note>"a certain point" corresponds to the center of the circle (blue point in the figure)</note>, and all <note>"straight lines" correspond to the radius in the problem (marked as r=3 in in the figure)</note>. The given radius of the circle in the problem is <note>r=3 inches</note>, which determines the size of the circle. In the figure, <note>all points that are 3 inches away from the center of the circle are on the circle</note>.

Definition of Radius

The original text: <note>The point is the center of the circle</note>, <note>The point is any point on the circle</note>, <note>The line segment is the line segment from the center of the circle to any point on the circle</note>, therefore <note>The line segment is the radius of the circle</note>. The problem gives <note>"Radius is 3 inches"</note>, i.e., <note>r=3 inches</note>, which determines the size of the circle.

Circumference Formula of Circle

<note>Point O is the center of the circle</note>, <note>line segments OA and OB are the radius r</note>. According to the circumference formula of the circle, the circumference C of the circle is equal to 2π multiplied by the radius r, that is, C=2πr, therefore <note>circumference C=6π inches</note>.

Arc Length Formula of a Circle

In the figure of this problem, <note>the central angle of the arc is 45°, i.e., m=45°</note>, so substituting it into the formula: <note>𝓁 = 6𝜋 × 45/360</note>, the simplified result is 3𝜋/4 inches.


Question:
What is the length of $\overline{SV}$ ?

SV= $\Box$

Answer: SV=26

1. Given arcs SV and TU on the circumference, let the center of the circle be O, and the corresponding central angles SOV and TOU are both 63°.

2. According to the central angle theorem, the central angles corresponding to arcs SV and TU are the same, indicating that the arcs are equal.

3. According to the central angle theorem, if two arcs are equal in the same circle, then their corresponding chords are also equal. Therefore, chords SV and TU are equal.

4. Given chord TU = 26, according to the conclusion in step 3, chord SV = chord TU = 26.

5. Through the above reasoning, the final answer is that the length of chord SV is 26.

Definition of Chord

<note>Point S and point V are any two points on the circle</note>, <note>the line segment SV connects these two points</note>, so <note>the line segment SV is a chord of the circle</note>. Similarly, <note>point T and point U are any two points on the circle</note>, <note>the line segment TU connects these two points</note>, so <note>the line segment TU is also a chord of the circle</note>.

Definition of Arc

<note>There are two points S and V on the circle</note>, arc SV is a segment of the curve connecting these two points. According to the definition of arc, <note>arc SV is a segment of the curve between two points S and V on the circle</note>. <note>There are two points T and U on the circle</note>, arc TU is a segment of the curve connecting these two points. According to the definition of arc, <note>arc TU is a segment of the curve between two points T and U on the circle</note>.

Property of Central Angle

In the figure of this problem, <note>the central angle subtended by arc SV is angle SOC</note>, <note>the degree measure of arc SV equals the degree measure of angle SOC</note>. <note>The central angle subtended by arc TU is angle TOU</note>, <note>the degree measure of arc TU equals the degree measure of angle TOU</note>.

Central Angle Theorem

In the figure of this problem, <note>angle SOV and angle TOU are two equal central angles within the same circle</note>, so <note>the corresponding arcs SV and TU are equal</note>, <note>the corresponding chords SV and TU are equal</note>.

<note>The arcs SV and TU within the same circle are equal</note>, then <note>the central angles SOV and TOU are equal</note>, <note>the chords SV and TU are equal</note>.


Question:
What is the length of $\overline{XY}$ ?

XY= $\Box$

Answer: XY=38

1. From the figure, we know arc XY = arc VW.

2. According to the central angle theorem, equal arcs on the circumference imply that XY and VW are equal chords, because in a circle, chords that subtend equal arcs are equal.

3. Given that the length of chord VW is 38, the length of chord XY is also 38.

4. Based on the above reasoning, the final answer is 38.

<note>The arcs XY and VW within the same circle are equal</note>, <note>the corresponding chords XY and VW are equal</note>, i.e., <note>chord XY = VW = 38</note>.

Definition of Central Angle

In the figure of this problem, in circle O, <note>points X, Y, V, and W are points on the circle</note>, <note>the center of the circle is point O</note>. <note>The angle ∠XOY formed by the lines OX and OY is called a central angle</note>. <note>The angle ∠VOW formed by the lines OV and OW is called a central angle</note>.

Definition of Arc

<note>There are two points X and Y on the circle</note>, <note>Arc XY is a segment of the curve connecting these two points</note>. Similarly, <note>there are two points V and W on the circle</note>, <note>Arc VW is a segment of the curve connecting these two points</note>. According to the definition of an arc, <note>Arc XY is a segment of the curve between the two points X and Y on the circle</note>, <note>Arc VW is a segment of the curve between the two points V and W on the circle</note>.

Definition of Chord

In the figure of this problem, in the circle, <note>points X and Y are any two points on the circle</note>, <note>the line segment XY connects these two points</note>, so <note>the line segment XY is a chord of the circle</note>. Similarly, <note>points V and W are any two points on the circle</note>, <note>the line segment VW connects these two points</note>, so <note>the line segment VW is a chord of the circle</note>.


Question:
Find the area of △UVW.

Write your answer as an integer or as a decimal rounded to the nearest tenth.
 $\Box$  yd ^ 2

Answer: 814.0 yd ^ 2

1. Given triangle △UVW, m ∠U = 27°, m ∠W = 37°, and side length UV = 49 yd.

2. According to the triangle angle sum theorem, the sum of the three interior angles of the triangle is 180°: m ∠U + m ∠W + m ∠V = 180°.

3. Substitute the known values: 27° + 37° + m ∠V = 180°, resulting in 64° + m ∠V = 180°.

4. By calculation: m ∠V = 180° - 64° = 116°.

5. Now knowing two interior angles m ∠W and m ∠U, and their opposite side length UV, apply the sine rule to solve for side length u: w/sin ∠W = u/sin ∠U.

6. Use the known values to calculate: 49/sin(37°) = u/sin(27°).

7. Look up the sine values: sin(37°) ≈ 0.6018, sin(27°) ≈ 0.4540.

8. Substitute the sine values into the equation: 49/0.6018 = u/0.4540.

10. Now knowing two side lengths and their included angle, apply the triangle area formula (using the sine function): Area = 1/2 * u * w * sin( ∠V).

11. Calculate the value of sin( ∠V), where m ∠V = 116°, resulting in sin(116°) ≈ 0.8988.

12. Substitute into the formula to calculate the area: Area = 1/2 * 36.9658 * 49 * 0.8988.

13. The calculation result is: Area ≈ 814.091 yd².

14. Round the result to the nearest tenth: Area = 814.1 yd².

n. Through the above reasoning, the final answer is 814.1 yd².

Triangle Angle Sum Theorem

The sum of the interior angles of any triangle is 180°.

In <note>triangle UVW</note>, <note>angle WVU, angle VWU, and angle WUV</note> are <note>the three interior angles of triangle UWV</note>. According to the Triangle Angle Sum Theorem, <note>angle WVU + angle VWU + angle WUV = 180°</note>.

Sine Theorem

In any triangle, the ratio of the length of each side to the sine of its opposite angle is constant and equal to the diameter of the circumcircle, i.e., \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2r = D \) (where \( r \) is the radius of the circumcircle, and \( D \) is its diameter).

In <note>any triangle UVW</note>, <note>sides UW, VW, and UV correspond to angles ∠V, ∠U, and ∠W respectively</note>. According to the <note>Sine Theorem</note>, the ratio of the lengths of the sides to the sine values of their opposite angles is equal and is equal to the diameter of the circumscribed circle, that is: <note>UW/sin(∠V)=VW/sin(∠U)=UV/sin(∠W) = 2r = D</note> (where r is the radius of the circumscribed circle and D is the diameter). Given <note>side length UV = 49 yd</note> and <note>opposite angle ∠W = 37°</note>, we can solve for <note>side length VW and its opposite angle ∠U = 27°</note>. That is: <note>49 / sin(37°) = VW / sin(27°)</note>, solving: <note>VW ≈ 49 * sin(27°) / sin(37°) ≈ 36.9658 yd</note>.

Triangle Area Formula (Using Sine Function)

The area \( S \) of any triangle can be expressed as \( S = \frac{1}{2} \cdot a \cdot b \cdot \sin(C) \), where \( a \) and \( b \) are the lengths of two sides, and \( C \) is the angle between these two sides.

<note>In the figure of this problem, in triangle UVW, side WV and side UV are u and w respectively, and angle WVU is the included angle C between these two sides</note>. According to the triangle area formula, the area S of triangle UVW can be expressed as S = (1/2) * a * b * sin(C), that is, <note>S = (1/2) * WV * UV * sin(WVU)</note>.


Question:
The radius of a circle is 6 inches. What is the length of a 180° arc?

Give the exact answer in simplest form.
 $\Box$ 𝜋/ inches

Answer: 6𝜋 inches

1. Given the radius of the circle is 6 inches, find the length of the 180° arc. According to the arc length formula, i.e., arc length l = C * (m/360), where C is the circumference of the circle and m is the central angle.

2. First, we need to find the circumference C of the circle. According to the circumference formula C = 2πr, where r is the radius of the circle. Substituting the given r = 6, we get C = 2π * 6.

3. Calculate C = 2π * 6, obtaining C = 12π. Therefore, the circumference of the circle is 12π inches.

4. The formula for arc length l is l = C * (m/360). Given the central angle m = 180°, substituting C = 12π and m = 180, we get l = 12π * (180/360).

5. Calculate l = 12π * (180/360), simplifying to l = 12π * (1/2).

6. Obtain l = 6π. Therefore, the length of this 180° arc is 6π inches.

n. Through the above reasoning, the final answer is 6π.

Definition of Radius

A radius is a line segment joining the center of a circle to any point on the circumference of the circle.

In the figure of this problem, in the circle, <note>the point is the center of the circle</note>, <note>the point is any point on the circle</note>, <note>the line segment is the line segment from the center of the circle to any point on the circle</note>, therefore <note>the line segment is the radius of the circle</note>. It is known that <note>the radius of the circle r = 6 inches</note>, that is, <note>the distance from the center of the circle to any point on the circle is 6 inches</note>.

Circumference Formula of Circle

<note>The point is the center of the circle</note>, <note>the segment is the radius r</note>. According to the circumference formula of a circle, the circumference C of the circle is equal to 2π multiplied by <note>radius r</note>, that is, C=2πr. Given <note>the radius of the circle r = 6 inches</note>, therefore <note>the circumference of the circle C = 2π * 6 = 12π inches</note>.

Definition of Central Angle

An angle formed by two radii connecting two points on the circumference of a circle to the center of the circle is called a central angle.

<note>The central angle of a circle is 180°</note>, which corresponds to the semicircular arc of the circle. The central angle is formed by <note>the angle composed of the lines connecting two points on the circle to the center of the circle</note>.

Arc Length Formula of a Circle

In the figure of this problem, to find <note>180° arc length l</note>, it is necessary to calculate <note>l = 12π * (180/360)</note> = 6π inches.

Definition of Angle Bisector

An angle bisector is a line that originates from the vertex of an angle and divides the angle into two congruent angles.

The <note>diameter of the circle</note> divides the circumference into two <note>equal arc lengths</note>, each <note>arc length corresponding to a central angle of 180°</note>.


Question:
What is the length of $\overline{BE}$ ?

BE= $\Box$

Answer: BE=48

1. As shown in the figure, let the center of the circle be O. There are arcs BE and CD in the circle, both of which are known to be 75°. According to the property of central angles, the corresponding central angles ∠BOE and ∠COD are equal, both being 75°.

2. According to the central angle theorem, chord BE = chord CD = 48

3. Through the above reasoning, the final answer is BE = 48.

Definition of Central Angle

<note>Point B and point E are two points on the circle</note>, <note>the center of the circle is point O</note>. <note>The angle ∠BOE formed by the lines OB and OE is called the central angle</note>. Similarly, <note>point C and point D are two points on the circle</note>, <note>the angle ∠COD formed by the lines OC and OD is called the central angle</note>. ∠BOE and ∠COD are both 75°.

Definition of Arc

In the diagram of this problem, there are two points <note>B</note> and <note>E</note> on the circle, and <note>arc BE</note> is a segment of the curve connecting these two points. Similarly, there are two points <note>C</note> and <note>D</note> on the circle, and <note>arc CD</note> is a segment of the curve connecting these two points. According to the definition of an arc, <note>arc BE</note> is a segment of the curve between the two points <note>B</note> and <note>E</note> on the circle, and <note>arc CD</note> is a segment of the curve between the two points <note>C</note> and <note>D</note> on the circle.

Definition of Chord

A chord is defined as a line segment that connects any two points on a circle.

<note>In a circle, points B and E are any two points on the circle</note>, <note>line segment BE connects these two points</note>, so <note>line segment BE is a chord of the circle</note>. Similarly, <note>points C and D are any two points on the circle</note>, <note>line segment CD connects these two points</note>, so <note>line segment CD is a chord of the circle</note>.

Property of Central Angle

The degree measure of an arc is equal to the degree measure of the central angle that subtends the arc.

In the figure of this problem, <note>the central angle corresponding to arc BE is angle BOE</note>, <note>the degree of arc CD equals the degree of angle COD</note>.

Central Angle Theorem

In the same circle or in congruent circles, arcs corresponding to equal central angles are equal, chords corresponding to equal central angles are equal, and the perpendicular distances from the center to these chords are also equal.

<note>Angle BOE and angle COD are two equal central angles in the same circle</note>, so <note>the corresponding arcs BE and CD are equal</note>, <note>the corresponding chords BE and CD are equal</note>.


Question:
Which two triangles are congruent by the AAS Theorem? Complete the congruence statement.

△ $\Box$ ≅△ $\Box$

Answer: △PQR≅△GEF

1. Observing the given figure, we need to find two sets of triangles where two angles and one non-included side of each triangle correspond respectively.

2. Based on the given conditions, first determine two pairs of equal angles: ∠Q ≅ ∠E and ∠P ≅ ∠G.

3. Next, find the sides that are not adjacent to the other two angles: it is known that QR ≅ EF.

4. According to the congruent triangles theorem (AAS), this theorem states that if two angles of one triangle are equal to the corresponding two angles of another triangle, and their non-included sides are equal to the corresponding non-included sides of the other triangle, then the two triangles are congruent.

5. Apply the congruent triangles theorem (AAS) and choose the two sets of triangles that meet the conditions: △PQR and △EFG.

6. Locate the corresponding vertices to complete the congruence statement: since ∠Q corresponds to ∠E, ∠P corresponds to ∠G, and side QR corresponds to side EF, R must correspond to F.

7. Finally, combining the above reasoning and matching relationships, the congruence statement of the triangles is: △PQR ≅ △GEF.

8. After the above reasoning, the final answer is △PQR ≅ △GEF.

Definition of an Angle

An angle is a geometric figure formed by two rays sharing a common endpoint. This common endpoint is called the vertex of the angle, and the two rays are referred to as the sides of the angle.

<note>∠PQR is an angle formed by rays PQ and QR with vertex Q</note>; <note>∠FEG is an angle formed by rays EF and EG with vertex E</note>; <note>∠P is an angle formed by rays PQ and PR with vertex P</note>; <note>∠G is an angle formed by rays FG and GE with vertex G</note>.

Definition of Congruent Triangles

Two triangles are congruent if and only if their corresponding sides are equal and their corresponding angles are equal.

In the diagram of this problem, triangles <note>PQR</note> and <note>GEF</note> are congruent triangles, meaning that the corresponding sides and angles of triangle <note>PQR</note> are equal to those of triangle <note>GEF</note>, specifically:
Side <note>PQ</note> = Side <note>GE</note>
Side <note>QR</note> = Side <note>EF</note>
Side <note>RP</note> = Side <note>FG</note>, and the corresponding angles are also equal:
Angle <note>PQR</note> = Angle <note>GEF</note>
Angle <note>QRP</note> = Angle <note>EFG</note>
Angle <note>RPQ</note> = Angle <note>FGE</note>.

Congruence Theorem for Triangles (AAS)

Two triangles are congruent if two angles and the included side of one triangle are respectively equal to two angles and the included side of the other triangle.

<note>△PQR and △GEF satisfy the AAS Congruence Theorem</note>, because <note>∠Q ≅ ∠E</note>, <note>∠P ≅ ∠G</note> and <note>QR ≅ EF</note>. Therefore, according to the theorem, <note>△PQR ≅ △GEF</note>.


Question:
What is the measure of $\overset{\frown}{UV}$ ?

m $\overset{\frown}{UV}$ = $\Box$ °

Answer: m \$\overset{\frown}{UV}\$ =53°

1. Observe the two chords inside the circle in the figure, let the central angle be O, the lengths of chord VU and chord WT are equal, each being 22 units.

2. According to the central angle theorem: In the same circle or congruent circles, if two chords are equal, then their corresponding central angles are equal, and the arcs they subtend are also equal.

3. The arcs corresponding to chord VU and chord WT are arc VU and arc WT, respectively. From step 2, it is known that arc VU = arc WT = 53°.

4. Through the above reasoning, the final answer is 53°.

Definition of Chord

In the figure of this problem, in the circle, <note>points V and W are any two points on the circle</note>, <note>the line segment VW connects these two points</note>, so <note>the line segment VW is a chord of the circle</note>. Similarly, <note>points U and T are any two points on the circle</note>, <note>the line segment UT connects these two points</note>, so <note>the line segment UT is a chord of the circle</note>.

Definition of Arc

In the figure of this problem, there are two points T and W on the circle, <note>arc TW</note> is a segment of the curve connecting these two points. Similarly, there are two points U and V on the circle, <note>arc UV</note> is a segment of the curve connecting these two points. According to the <note>definition of arc</note>, arc TW is a segment of the curve between two points T and W on the circle, and arc UV is a segment of the curve between two points U and V on the circle.

Central Angle Theorem

<note>Chords WT and VU within the same circle are equal</note>, then <note>the central angles WOT and VOU subtended by them are equal</note>, and the arcs <note>WT and VU subtended by them are equal</note>.


Question:
The radius of a circle is 1 meter. What is the length of a 135° arc?

Give the exact answer in simplest form.
 $\Box$ 𝜋/ meters

Answer: 3/4𝜋 meters

1. Let the center of the circle be O, and the 135° angle be ∠AOB. According to the given conditions, the radius of the circle is 1 meter.

2. According to the formula for the circumference of a circle C = 2πr, where r is the radius of the circle. Substituting r = 1, we get C = 2π × 1 = 2π meters.

3. The length of the arc is related to the circumference of the circle and the degree of the central angle. According to the formula for the length of an arc 𝓁 = C × (m/360), where m is the degree of the central angle.

4. Substituting the given central angle m = 135° and the previously calculated circumference C = 2π meters into the arc length formula, we get 𝓁 = 2π × (135/360).

5. Calculating 2π × (135/360), equals 2π × (3/8) = 3π/4.

6. Through the above reasoning, the final answer is 3π/4 meters.

Definition of Radius

In the figure of this problem, in the circle, <note>point O is the center of the circle</note>, <note>point A is any point on the circle</note>, <note>line segment OA is the line segment from the center of the circle to any point on the circle</note>, therefore <note>line segment OA is the radius of the circle</note>, with a length of 1 meter.

Circumference Formula of Circle

In the figure of this problem, <note>in the circle, point O is the center</note>, <note>line segment OA is the radius r</note>. According to the <note>circumference formula of the circle</note>, the circumference C of the circle is equal to 2π multiplied by <note>radius r</note>, that is, <note>C=2πr</note>, where <note>r=1 meter</note>, therefore <note>C=2π×1=2π meters</note>.

Definition of Central Angle

The center of the circle is point <note>O</note>, the angle ∠AOB formed by the lines <note>OA and OB from points A and B on the circle to the center O</note> is called the <note>central angle</note>, and <note>this central angle is 135°</note>.

Arc Length Formula of a Circle

Arc Length <note>弧长𝓁</note> = 2π × (135/360), calculated as <note>𝓁 = 3π/4 meters</note>.

Definition of Arc

In the figure of this problem, there are two points <note>A and B</note> on the circle, and <note>arc AB</note> is a segment of the curve connecting these two points. According to the definition of an arc, arc AB is a segment of the curve between the two points <note>A and B</note> on the circle.


Question:
What is the length of $\overline{BC}$ ?

BC= $\Box$

Answer: BC=36

1. Given arc DE=74°, arc BC=74°, DE and BC are chords of the circle, and the corresponding arcs are equal.

2. According to the central angle theorem, if two chords in a circle subtend equal arcs, then the two chords are equal. In this problem, chord DE and chord BC subtend the same arc and the inscribed angles are equal, so we conclude DE=BC.

3. Given chord DE=36, so according to the conclusion from the previous step, chord BC=36.

4. Through the above reasoning, the final answer is 36.

Definition of Inscribed Angle

An inscribed angle is an angle whose vertex lies on the circumference of a circle and whose sides intersect the circle at two points.

<note>In the figure of this problem, in circle O, the vertex C of angle DCE is on the circumference</note>, and the two sides of angle DCE intersect <note>circle O at points D and E</note>. Therefore, angle DCE is an inscribed angle. Similarly, <note>the vertex D of angle BDC is on the circumference</note>, and the two sides of angle BDC intersect <note>circle O at points C and B</note>. Therefore, angle BDC is also an inscribed angle.

Definition of Chord

In the problem diagram, in the circle, <note>points D and E are any two points on the circle</note>, <note>line segment DE connects these two points</note>, so <note>line segment DE is a chord of the circle</note>, with a length of 36; similarly, <note>points B and C are any two points on the circle</note>, <note>line segment BC connects these two points</note>, so <note>line segment BC is a chord of the circle</note>.

Central Angle Theorem

In the diagram of this problem, <note>angle DCE</note> and <note>angle BDC</note> are equal central angles in the same circle, so the corresponding <note>arc DE</note> and <note>arc BC</note> are equal, and the corresponding <note>chord DE</note> and <note>chord BC</note> are equal.


Question:
Find TU and the area of △UVW.

Write your answer as an integer or as a decimal rounded to the nearest tenth.
TU= $\Box$  mmArea= $\Box$  mm ^ 2

Answer: TU=13.1 mmArea=314.4 mm ^ 2

1. Given △TUW is a right triangle, where ∠T is a right angle. According to the definition of trigonometric functions, for any non-right angle in a right triangle, the sine function value equals the ratio of the opposite side to the hypotenuse.

2. In △TUW, given ∠W = 25°, UW = 31 mm. According to the definition of the sine function, sin(W) = opposite side TU / hypotenuse UW, i.e., sin(25°) = TU / 31.

3. Calculate sin(25°), obtaining sin(25°) ≈ 0.4226.

5. Determine the area of △UVW. Given VW = 48 mm, TU is the height of △UVW.

6. According to the triangle area formula: Area = 1/2 × base × height, we get:

7. The area of △UVW = 1/2 × VW × TU ≈ 1/2 × 48 × 13.1 = 314.4 square mm.

8. Through the above reasoning, the final answer is TU = 13.1 mm, and the area of △UVW is 314.4 mm².

Definition of Right Triangle

A triangle is classified as a right triangle if it has one interior angle measuring exactly 90 degrees.

In the figure of this problem, in triangle TUW, <note>angle ∠UTW is a right angle (90 degrees)</note>, so triangle TUW is a right triangle. <note>Side TU and side TW are the legs</note>, <note>side UW is the hypotenuse</note>.

Definition of Sine Function

In a right-angled triangle, the sine of an acute angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse.

In the figure of this problem, in the right triangle △TUW, <note>angle ∠W is an acute angle</note>, <note>side TU is the opposite side of angle ∠W</note>, <note>side UW is the hypotenuse</note>. According to the definition of the sine function, the sine value of angle ∠W is equal to the ratio of the opposite side TU to the hypotenuse UW, that is, <note>sin(∠W) = TU / UW</note>.

Area Formula of a Triangle

The area of any triangle is equal to its base multiplied by its height, divided by 2.

In the triangle UVW, the side VW is the base, and the segment UT is the height on this base, so the area of the triangle UVW is equal to the base VW multiplied by the height UT divided by 2, i.e., area = <note>(VW* UT) / 2</note>.


Question:
Find the area of △XYZ.

Write your answer as an integer or as a decimal rounded to the nearest tenth.
 $\Box$  mm ^ 2

Answer: 63.4 mm ^ 2

1. Given ∠Z=23° and ∠Y=28°, through the triangle angle sum theorem (i.e., the sum of the three interior angles of a triangle is 180°), ∠X can be found.

2. The triangle angle sum theorem states: ∠X + ∠Y + ∠Z = 180°.

3. Substitute the known angles: ∠X + 28° + 23° = 180°.

4. Calculate to get ∠X = 180° - 28° - 23° = 129°.

5. Let the side opposite ∠Y be y, the side opposite ∠X be x, and the side opposite ∠Z be z. Now, given y=14mm, ∠Y=28° and ∠Z=23°, the value of z can be found using the sine rule.

6. The sine rule states: y/sin(∠Y) = z/sin(∠Z).

7. Substitute the known side length and angles into the sine rule: 14/sin(28°) = z/sin(23°).

8. Solve the equation to get z = [14 * sin(23°)] / sin(28°).

10. Now, given z ≈ 11.65 mm, y = 14 mm and ∠X = 129°, the area can be calculated using the triangle area formula (using the sine function).

11. Triangle area: S = (1/2) * z * y * sin(∠X).

12. Substitute the known values into the area formula: S = (1/2) * 11.65 * 14 * sin(129°).

14. Through the above reasoning, the final answer is 63.4 mm^2.

Triangle Angle Sum Theorem

In <note>triangle XYZ</note>, <note>angle X, angle Y, and angle Z are the three interior angles of triangle XYZ</note>. According to the Triangle Angle Sum Theorem, <note>angle X + angle Y + angle Z = 180°</note>.

Sine Theorem

<note>In the figure of this problem, in any triangle XYZ, sides YZ, XY, and XZ correspond to angles ∠X, ∠Y, and ∠Z respectively</note>. According to the Sine Theorem, the ratio of the lengths of the sides to the sine values of their opposite angles is equal and is equal to the diameter of the circumscribed circle, that is: <note>YZ/sin(∠X)=XY/sin(∠Y)=XZ/sin(∠Z) = 2r = D</note> (where r is the radius of the circumscribed circle, D is the diameter). Through <note>14/sin(28°) = z/sin(23°)</note>, it is found that z ≈ 11.65 mm.

In the figure of this problem, in triangle XYZ, <note>side XY</note> and <note>side YZ</note> are respectively <note>11.65 mm and 14 mm</note>, <note>angle ∠XYZ is the included angle between these two sides, 129°</note>. According to the triangle area formula, the area S of triangle XYZ can be expressed as <note>S = (1/2) * 11.65 * 14 * sin(129°)</note>, that is <note>S = (1/2) * 11.65 * 14 * sin(129°)</note>.


Question:
Find the area of △VWX.

Write your answer as an integer or as a decimal rounded to the nearest tenth.
 $\Box$  m ^ 2

Answer: 11.5 m ^ 2

1. Given: m∠V = 26°, m∠X = 120°, and side XW = 9m. According to the triangle angle sum theorem, the sum of the interior angles of a triangle is 180°.

2. Let m∠W be the unknown angle, then m∠V + m∠X + m∠W = 180°. Substituting the given angles: 26° + 120° + m∠W = 180°.

3. Calculating: 146° + m∠W = 180°, rearranging gives m∠W = 180° - 146° = 34°.

4. According to the sine rule, the ratio of the lengths of sides to the sines of their opposite angles is equal in a triangle. Therefore, XW/sin(m∠V) = VW/sin(m∠X). Substituting the given values: XW/sin(26°) = 9/sin(120°).

5. Calculating the sine values: sin(26°) ≈ 0.4383, sin(120°) ≈ 0.8660. Substituting into the equation: XW/0.4383 = 9/0.8660.

6. Solving for XW by cross-multiplication: XW ≈ 9 * 0.4383 / 0.8660 ≈ 4.5550, thus side XW = 4.5550 m.

7. Now that we have sides VW, XW and their included angle m∠W = 34°, we can use the triangle area formula to calculate the area: Area = 1/2 * VW * XW * sin(m∠W).

8. Substituting the known values and calculated results, Area = 1/2 * 4.5550 * 9 * sin(34°).

9. Calculating sin(34°) ≈ 0.5591, Area = 1/2 * 4.5550 * 9 * 0.5591 ≈ 11.4601.

10. Rounding to one decimal place, the area of triangle △VWX is approximately 11.5 square meters.

Definition of Triangle

A geometric figure is called a triangle if it is formed by three non-collinear points and the line segments connecting them.

In the figure of this problem, <note>the triangle △VWX is a geometric figure composed of three non-collinear points V, W, X and their connecting line segments VW, WX, XV</note>. <note>Points V, W, X are the three vertices of the triangle</note>, <note>and the line segments VW, WX, XV are the three sides of the triangle</note>.

In the <note>triangle △VWX</note>, <note>side VW and side XW are a and b respectively</note>, and <note>angle ∠VWX is the included angle C between these two sides</note>. According to the triangle area formula, <note>the area S of triangle △VWX can be expressed as S = (1/2) * a * b * sin(C)</note>, which is <note>S = (1/2) * VW * XW * sin(∠VWX)</note>.

Triangle Angle Sum Theorem

The sum of the interior angles of any triangle is 180°.

<note>In triangle VWX, angle V, angle X, and angle W are the three interior angles of triangle VWX</note>, according to the Triangle Angle Sum Theorem, <note>angle V + angle X + angle W = 180°</note>.

Sine Theorem

In any triangle, the ratio of the length of each side to the sine of its opposite angle is constant and equal to the diameter of the circumcircle, i.e., \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2r = D \) (where \( r \) is the radius of the circumcircle, and \( D \) is its diameter).

<note>In the figure of this problem, in any triangle VWX, sides XW, VW, and VX correspond to angles ∠V, ∠X, and ∠W respectively</note>. According to the Sine Theorem, <note>the ratio of the length of each side to the sine of its opposite angle is equal and equals the diameter of the circumscribed circle</note>, that is: <note>XW/sin(∠V)=VW/sin(∠X)=VX/sin(∠W) = 2r = D (where r is the radius of the circumscribed circle, D is the diameter)</note>.


Question:
What is the length of $\overline{TW}$ ?

TW= $\Box$

Answer: TW=16

1. Draw the center point O, connect OU, OV, OT, OW. According to the definition of central angles, ∠UOV and ∠TOW are central angles. From the figure, we know that arc TW and arc UV are equal. According to the properties of central angles, we get ∠UOV=∠TOW. It is known that segments TW and UV are chords on the same circle.

2. According to the theorem of central angles, if two arcs are equal, then the chord lengths are also equal. Therefore, segments TW and UV are equal.

3. In the problem, it is already marked that segment UV= 16.

4. Therefore, according to the conclusion in step 2, TW=UV= 16.

5. Through the above reasoning, the final answer is 16.

Properties of Central Angles

The measure of a central angle is equal to the measure of the arc that it intercepts.

The arc corresponding to <note>central angle ∠UOV</note> is <note>arc UV</note>. According to the properties of central angles, the degree measure of a central angle is equal to the degree measure of the corresponding <note>arc UV</note>. Similarly, the degree measure of the central angle <note>∠TOW</note> is equal to the degree measure of the corresponding <note>arc TW</note>. From the diagram, it is known that <note>arc TW</note> is equal to <note>arc UV</note>, therefore <note>∠UOV=∠TOW</note>.

Central Angle Theorem

In the diagram of this problem, <note>the arcs UV and TW within the same circle are equal</note>, then <note>the central angles UOV and TOW are equal</note>, <note>the chords UV and TW are equal</note>.

Definition of Central Angle

<note>In circle O, points U, V, T, W are points on the circle, and the center of the circle is point O</note>. The angle ∠UOV formed by the lines <note>OU and OV is called the central angle</note>. The angle ∠TOW formed by the lines <note>OT and OW is called the central angle</note>.

Definition of Arc

An arc is a curve on a circle that is defined by two distinct points on the circle.

In the figure of this problem, there are two points <note>T and W</note> on the circle, and <note>arc TW</note> is a curve connecting these two points. There are two points <note>U and V</note> on the circle, and <note>arc UV</note> is a curve connecting these two points.

Definition of Chord

In the figure of this problem, in the circle, <note>points T and W are any two points on the circle</note>, <note>the segment TW connects these two points</note>, so <note>the segment TW is a chord of the circle</note>. <note>Points U and V are any two points on the circle</note>, <note>the segment UV connects these two points</note>, so <note>the segment UV is a chord of the circle</note>.


Question:
Find WY and the area of △VWX.

Write your answer as an integer or as a decimal rounded to the nearest tenth.
WY= $\Box$  mmArea= $\Box$  mm ^ 2

Answer: WY=13.1 mmArea=91.7 mm ^ 2

1. From the figure in the problem, it is known that in △VWY, ∠VYW is 90°, so △VWY is a right triangle.

2. Using the definition of the sine function: In a right triangle, sin(θ) = opposite side / hypotenuse.

3. In △VWY, it is known that ∠WVY = 55°, the hypotenuse VW = 16 mm, and we need to find the opposite side WY.

4. According to the definition of sine, sin(55°) = WY / 16.

5. Substituting sin(55°) ≈ 0.8191, we get 0.8191 = WY / 16.

6. By rearranging, we get WY = 0.8191 × 16.

7. Calculating, we get WY ≈ 13.1 mm.

8. Now calculate the area of △VWX. Since YW is the height in △VWX, the base of △VWX is VX = 14 mm, and the height is WY = 13.1 mm.

9. Using the triangle area formula: Area = 1/2 × base × height.

10. Substituting the base as 14 mm and the height as 13.1 mm, we get Area = 1/2 × 14 × 13.1.

11. Calculating, we get Area ≈ 91.7 mm².

12. Through the above reasoning, the final answer is WY ≈ 13.1 mm, Area ≈ 91.7 mm².

Definition of Sine Function

In the figure of this problem, in the right triangle <note>△VWY</note>, <note>∠WVY is an acute angle</note>, <note>side WY is the opposite side of ∠WVY</note>, <note>side VW is the hypotenuse</note>. According to the definition of the sine function, <note>the sine value of angle ∠WVY is equal to the ratio of the opposite side WY to the hypotenuse VW</note>, that is, <note>sin(55°) = WY / VW</note>.

Definition of Right Triangle

In the figure of this problem, in triangle VWY, <note>angle VYW is a right angle (90 degrees)</note>, therefore <note>triangle VYW is a right triangle</note>.


Question:
Find WZ and the area of △WXY.

Write your answer as an integer or as a decimal rounded to the nearest tenth.
WZ= $\Box$  mArea= $\Box$  m ^ 2

Answer: WZ=3.7 mArea=22.2 m ^ 2

1. Given that angle WZY is a right angle in the right triangle, WY = 6 meters, angle Y = 38°.

2. According to the definition of the sine function, in the right triangle WYZ, sin(angle Y) = opposite/hypotenuse, i.e., sin(Y) = WZ/WY.

3. Substituting the given conditions, sin(38°) = WZ/6, and according to calculations, sin(38°) is approximately 0.6157.

4. Solving the equation WZ/6 = 0.6157, we get WZ ≈ 3.6942 meters.

5. We continue to find the area of △WXY. The base of this triangle is XY = 12 meters, and the height is WZ.

6. According to the area formula of a triangle, Area = 1/2 * base * height, i.e., Area = 1/2 * XY * WZ.

7. Substituting the specific values, we get Area = 1/2 * 12 * 3.6942.

8. Calculating, we get Area ≈ 22.1652 square meters.

9. Rounding the area result to one decimal place, we get Area ≈ 22.2 square meters.

10. Through the above reasoning, the final answer is WZ ≈ 3.7 meters, and the area of △WXY is approximately 22.2 square meters.

<note>Angle WZY is a right angle (90 degrees)</note>, therefore triangle WYZ is a right triangle. <note>Side WZ and side YZ are the legs</note>, <note>side WY is the hypotenuse</note>.

Definition of Sine Function

<note>In the figure of this problem, in the right triangle WYZ, angle WYZ is an acute angle</note>, <note>side WZ is the opposite side of angle WYZ</note>, <note>side WY is the hypotenuse</note>. According to the definition of the sine function, the sine value of angle WYZ is equal to the ratio of the opposite side WZ to the hypotenuse WY, that is, <note>sin(WYZ) = WZ / WY</note>.

Area Formula of a Triangle

In <note>triangle WXY</note>, <note>side XY is the base</note>, <note>segment WZ is the height from the base</note>, so the <note>area of triangle WXY</note> is equal to <note>base XY multiplied by height WZ divided by 2</note>, i.e., area = <note>(XY * WZ) / 2</note>.


Question:
Find the area of △VWX.

Write your answer as an integer or as a decimal rounded to the nearest tenth.
 $\Box$  yd ^ 2

Answer: 296.5 yd ^ 2

1. Given that \( m \angle W = 21^\circ \), \( m \angle V = 121^\circ \), and side \( WX = 48 \) yd. To find the area of triangle \( \triangle VWX \), we need two side lengths and the included angle.

2. First, determine the degree of \( \angle X \). According to the triangle angle sum theorem, the sum of the three interior angles of a triangle is \( 180^\circ \).

3. Let \( m \angle X = x \). According to the equation \( m \angle W + m \angle V + m \angle X = 180^\circ \), we have \( 21^\circ + 121^\circ + x = 180^\circ \).

4. Calculating, we get \( 142^\circ + x = 180^\circ \), thus \( x = 38^\circ \), i.e., \( m \angle X = 38^\circ \).

5. Now use the sine rule to find the length of side \( w \) opposite \( \angle W \). The formula is \( \frac{v}{\sin(V)} = \frac{w}{\sin(W)} \).

6. Substitute the known values, \( \frac{48}{\sin 121^\circ} = \frac{w}{\sin 21^\circ} \).

7. Calculate \( \sin 121^\circ \approx 0.8571 \) and \( \sin 21^\circ \approx 0.3583 \), obtaining \( \frac{48}{0.8571} = \frac{w}{0.3583} \).

8. Solving, we get \( 55.9984 = \frac{w}{0.3583} \), thus \( w \approx 20.066 \) yd.

9. Given the two side lengths \( w \approx 20.066 \) yd, \( v = 48 \) yd, and the included angle \( m \angle X = 38^\circ \), apply the triangle area formula \( \text{Area} = \frac{1}{2}ab\sin(C) \).

10. Substitute the corresponding values: Area \( = \frac{1}{2} \times 20.066 \times 48 \times \sin 38^\circ \).

11. Calculating, we get \( \sin 38^\circ \approx 0.6156 \), so the area \( \approx \frac{1}{2} \times 20.066 \times 48 \times 0.6156 = 296.4631 \text{ yd}^2 \).

12. Round the obtained area to one decimal place, resulting in \( 296.5 \text{ yd}^2 \).

Through the above reasoning, the final answer is 296.5 \text{ yd}^2.

Triangle Angle Sum Theorem

In the diagram of this problem, in triangle \(\triangle VWX\), <note>\(\angle W\), \(\angle V\) and \(\angle X\) are the three interior angles of triangle \(\triangle VWX\)</note>, according to the Triangle Angle Sum Theorem, <note>\(\angle W + \angle V + \angle X = 180^\circ\)</note>.

Sine Theorem

<note>In the figure of this problem</note>, in triangle <note>WVX</note>, <note>sides WV, VX, and WX correspond to angles X, W, and V respectively</note>. According to the Sine Theorem, the ratio of the lengths of each side to the sine of its opposite angle is equal and is equal to the diameter of the circumscribed circle, that is: <note>WX/sin(V)=WV/sin(X)=VX/sin(W)</note> = 2r = D (<note>where r is the radius of the circumscribed circle, D is the diameter</note>).

The area \( S \) of any triangle can be expressed as \( S = \frac{1}{2} \cdot a \cdot b \cdot \sin(C) \), where \( a \) and \( b \) are the lengths of two sides, and \( C \) is the angle between these two sides.

In the diagram of this problem, in triangle <note>WVX</note>, side <note>VX</note> and side <note>WX</note> are <note>a</note> and <note>b</note> respectively, and angle <note>X</note> is the included angle <note>C</note> between these two sides. According to the triangle area formula, the area <note>S</note> of triangle <note>WVX</note> can be expressed as <note>S = (1/2) * a * b * sin(C)</note>, that is, <note>S = (1/2) * WX * VX * sin(X)</note>.


Question:
Find TU and the area of △UVW.

Write your answer as an integer or as a decimal rounded to the nearest tenth.
TU= $\Box$  kmArea= $\Box$  km ^ 2

Answer: TU=10.6 kmArea=243.8 km ^ 2

1. Given △TUW is a right triangle with the right angle at ∠WTU, UW = 20 km, ∠W = 32°.

2. Since ∠W is an angle of the right triangle and not the right angle, we can use the sine function definition: sin(∠W) = opposite / hypotenuse = TU / UW.

3. Substitute the given values sin(32°) = TU / 20.

4. Using a table or calculator, we get sin(32°) ≈ 0.5299.

5. Therefore, TU = 20 * sin(32°) ≈ 20 * 0.5299 = 10.598 km.

6. Calculate the area of △UVW, given VW = 46 km, TU is the height of the triangle.

7. Using the triangle area formula: Area = 1/2 * base * height.

8. Substitute the given values: Area = 1/2 * 46 * 10.598 ≈ 243.8 km².

9. Through the above reasoning, the final answer is TU = 10.6 km and the area of △UVW is 243.8 km².

Definition of Sine Function

In the <note>right triangle TUW</note>, <note>∠TWU is an acute angle</note>, side <note>TU is the opposite side of angle ∠TWU</note>, side <note>UW is the hypotenuse</note>. According to the definition of the sine function, the <note>sine value of angle ∠TUW</note> is equal to the ratio of the opposite side <note>TU to the hypotenuse UW</note>, that is, sin(32°) = <note>TU / UW</note>.

Definition of Right Triangle

In the figure of this problem, in triangle TUW, <note>angle ∠UTW is a right angle (90 degrees)</note>, so triangle TUW is a right triangle. <note>Side TU and side TW are the legs</note>, <note>side UW is the hypotenuse</note>.


Question:
Find VY and the area of △WXY.

Write your answer as an integer or as a decimal rounded to the nearest tenth.
VY= $\Box$  kmArea= $\Box$  km ^ 2

Answer: VY=2.4 kmArea=14.4 km ^ 2

1. Given ∠VXY = 90°, it indicates that △VXY is a right triangle.

2. In the right triangle △VXY, given ∠XYV = 20°, XY = 7 km. According to the definition of sine, sin ∠VXY = opposite side (VY) / hypotenuse (XY).

3. Substitute the given conditions to calculate, sin 20° = VY / 7.

4. Look up or calculate sin 20° ≈ 0.3420, so 0.3420 = VY / 7.

5. Rearrange to solve for VY, VY = 7 × 0.3420 ≈ 2.394 km.

6. Calculate the area of △WXY. Given WX = 12 km, so area = 1/2 × WX × VY.

7. Substitute the values, area = 1/2 × 12 × 2.394 = 14.364 km².

8. Round the area result to 1 decimal place to get 14.4 km².

9. Through the above reasoning, the final answer is VY = 2.4 km, Area = 14.4 km².

In the figure of this problem, in triangle VXY, <note>angle VXY is a right angle (90 degrees)</note>, therefore <note>triangle VXY is a right triangle</note>. <note>Side VY and side VX are the legs</note>, <note>side XY is the hypotenuse</note>.

Definition of Sine Function

In the diagram of this problem, in the right triangle VXY, <note>angle ∠XYV is an acute angle</note>, <note>side VY is the opposite side of angle ∠XYV</note>, <note>side XY is the hypotenuse</note>. According to the definition of the sine function, <note>the sine value of angle ∠YXV is equal to the ratio of the opposite side VY to the hypotenuse XY, that is, sin(∠YXV) = VY / XY</note>.

Area Formula of a Triangle

<note>Side WX is the base</note>, <note>Segment YV is the height on this base</note>, so the area of triangle WXY is equal to the base WX multiplied by the height YV divided by 2, i.e., Area = <note>(WX * YV) / 2</note>.


Question:
Find the area of △TUV.

Write your answer as an integer or as a decimal rounded to the nearest tenth.
 $\Box$  cm ^ 2

Answer: 16.4 cm ^ 2

1. Given m∠V=20°, m∠U=113°, and the length of side TV is 11 cm.

2. According to the triangle angle sum theorem, the sum of the interior angles of a triangle is 180°, that is, m∠V + m∠U + m∠T = 180°.

3. Substitute the given angles into the formula, 20° + 113° + m∠T = 180°, to get m∠T = 180° - 133° = 47°.

4. After determining m∠T=47°, the three interior angles of triangle △TUV are m∠V=20°, m∠U=113°, and m∠T=47°.

5. Using the sine rule: a/sin(A) = b/sin(B) = c/sin(C). In this problem, given TV=11 cm (corresponding to m∠U=113°), TU=unknown (corresponding to m∠V=20°). Applying the sine rule, we get TU/sin(20°) = 11/sin(113°).

6. Calculate sin(113°) ≈ 0.9205 and sin(20°) ≈ 0.3420, substitute them into the sine rule formula, TU/0.3420 ≈ 11/0.9205.

7. Solving the above equation gives: TU ≈ 11 * 0.3420 / 0.9205 ≈ 4.0869 cm.

8. Now we know the length of side TU ≈ 4.0871 cm, the length of side TV = 11 cm, and the included angle m∠T=47°.

9. According to the triangle area formula (using the sine function): Area = 1/2 * TU * TV * sin(T).

10. Substitute the known values into the formula to get: Area = 1/2 * 4.0869 * 11 * sin(47°).

11. Calculate sin(47°) ≈ 0.7314, further get Area ≈ 1/2 * 4.0869 * 11 * 0.7314 ≈ 16.4404 cm².

12. Finally, rounding the area to one decimal place, the area of △TUV is 16.4 cm².

In the diagram of this problem, in triangle TUV, side <note>TU</note> and side <note>TV</note> are <note>a</note> and <note>b</note> respectively, and angle <note>UTV</note> is the included angle between these two sides <note>C</note>. According to the triangle area formula, the area S of triangle TUV can be expressed as S = (1/2) * a * b * sin(C), that is, S = (1/2) * <note>TU</note> * <note>TV</note> * sin(<note>UTV</note>).

Triangle Angle Sum Theorem

In the figure of this problem, in triangle <note>TUV</note>, <note>angle V</note>, <note>angle U</note>, and <note>angle T</note> are the three interior angles of triangle TUV. According to the Triangle Angle Sum Theorem, <note>angle V + angle U + angle T = 180°</note>.

Sine Theorem

In the diagram of this problem, in any triangle <note>△TUV</note>, <note>sides TU, TV, and VU correspond to angles ∠V, ∠U, and ∠T respectively</note>. According to the Sine Theorem, <note>the ratio of the length of each side to the sine of its opposite angle is equal and equals the diameter of the circumscribed circle</note>, that is: <note>TU/sin(∠V)=TV/sin(∠U)=VU/sin(∠T) = 2r = D (where r is the radius of the circumscribed circle, D is the diameter)</note>.


Question:
Find the area of △TUV.

Write your answer as an integer or as a decimal rounded to the nearest tenth.
 $\Box$  m ^ 2

Answer: 85.0 m ^ 2

1. Given m∠V = 129°, m∠U = 24°, and side TV = 14 m.

2. We need to calculate m∠T. According to the triangle angle sum theorem, the sum of the interior angles of a triangle is 180°.

3. From m∠V + m∠U + m∠T = 180°, substituting the given angles, we get 129° + 24° + m∠T = 180°.

4. Solving the equation 129° + 24° + m∠T = 180°, we get m∠T = 27°.

5. Next, we use the sine rule to solve for the length of side UT, which is opposite to angle V. The sine rule is a/sin(A) = b/sin(B) = c/sin(C) and applies to any triangle ABC.

6. For triangle TUV, given the length of side TV (14 m) and corresponding angle U (24°), as well as angle V (129°), using the sine rule TV/sin(U) = UT/sin(V), i.e., 14/sin(24°) = UT/sin(129°).

7. Calculating, we get sin(24°) ≈ 0.4067, sin(129°) ≈ 0.7771.

8. Substituting the values into the equation, 14/0.4067 = UT/0.7771, solving this equation we get UT ≈ 26.7504 m.

9. Now we know two sides TV = 14 m, UT ≈ 26.7504 m, and the included angle ∠T = 27°.

10. Using the triangle area formula (using the sine function), for known two sides and included angle: Area = 1/2·ab·sin(C), i.e., Area = 1/2·TV·UT·sin(T).

11. Substituting the three values Area = 1/2·14·26.7504·sin(27°), where sin(27°) ≈ 0.4540.

12. Calculating the area Area = 1/2·14·26.7504·0.4540 ≈ 85.0128 m².

13. Rounding to the nearest tenth, the final calculated area of triangle TUV is approximately 85.0 square meters.

Triangle Angle Sum Theorem

<note>In triangle TUV, angles ∠T, ∠U, and ∠V are the three interior angles of triangle TUV</note>, according to the Triangle Angle Sum Theorem, <note>∠T + ∠U + ∠V = 180°</note>.

Sine Theorem

In the <note>triangle TUV</note>, the sides <note>TU, TV, and UV</note> correspond to the angles <note>∠V, ∠U, and ∠T</note> respectively. According to the <note>Sine Theorem</note>, the ratio of the lengths of the sides to the sine of their opposite angles is equal and is also equal to the diameter of the circumscribed circle, that is: <note>VT/sin(∠U) = UT/sin(∠V) = UV/sin(∠T) = 2r = D (where r is the radius of the circumscribed circle, D is the diameter)</note>. Given <note>TV = 14 m, ∠U = 24°, ∠V = 129°</note>, the length of the side <note>UT</note> can be determined.

In the diagram of this problem, in triangle <note>TUV</note>, side <note>VT</note> and side <note>TU</note> are 14 m and 26.7504 m respectively, and <note>∠T</note> is the angle between these two sides, 27°. According to the triangle area formula, the area <note>S</note> of triangle <note>TUV</note> can be expressed as <note>S = (1/2) * a * b * sin(C)</note>, that is, <note>S = (1/2) * 14 * 26.7504 * sin(27°)</note>.


Question:
Find TV and the area of △UVW.

Write your answer as an integer or as a decimal rounded to the nearest tenth.
TV= $\Box$  mArea= $\Box$  m ^ 2

Answer: TV=5.7 mArea=51.3 m ^ 2

1. Given △TUV is a right triangle, where ∠UTV is 90°, ∠VUT is 26°, and the side length UV=13 meters.

2. According to the definition of the sine function, sin(θ) represents the ratio of the opposite side to the hypotenuse. Therefore, for △TUV, we have sin(∠VUT)=TV/UV.

3. Substituting the known information, sin(26°)=TV/13.

4. Calculating sin(26°)=0.4384 (rounded to four decimal places), we obtain 0.4384=TV/13.

5. Solving the equation 0.4384=TV/13, we get TV=0.4384×13=5.6992 meters.

6. Therefore, TV rounded to one decimal place is 5.7 meters.

7. Now, calculate the area of △UVW. We consider UW as the base of the triangle and TV as the height. So, the area A of △UVW is: A=1/2×UW×TV.

8. Given UW=18 meters and TV=5.7 meters, the area A=1/2×18×5.7=51.3 square meters.

9. Through the above reasoning, the final answer is TV=5.7 meters and the area of △UVW is 51.3 square meters.

Definition of Right Triangle

In the diagram of this problem, <note>in triangle TUV, angle VTU is a right angle (90 degrees)</note>, therefore <note>triangle TUV is a right triangle</note>. <note>Side TU and side TV are the legs, and side UV is the hypotenuse</note>.

Definition of Sine Function

In the figure of this problem, in <note>right triangle TUV</note>, <note>angle ∠VUT is an acute angle</note>, <note>side TV is the opposite side of angle ∠VUT</note>, <note>side UV is the hypotenuse</note>. According to the definition of the sine function, <note>the sine value of angle ∠VUT is equal to the ratio of the opposite side TV to the hypotenuse UV, i.e., sin(∠VUT) = TV / UV</note>.

Area Formula of a Triangle

In the figure of this problem, in triangle UVW, <note>side UW is the base</note>, <note>segment TV is the height on this base</note>, so the area of triangle UVW is equal to <note>base UW multiplied by height TV divided by 2</note>, that is, area = <note>(UW * TV) / 2</note>.


Question:
Find VY and the area of △WXY.

Write your answer as an integer or as a decimal rounded to the nearest tenth.
VY= $\Box$  mmArea= $\Box$  mm ^ 2

Answer: VY=31.3 mmArea=438.2 mm ^ 2

1. Given △VXY is a right triangle, and ∠VXY=67°, the length of side XY is 34 mm.

2. To facilitate the calculation of the length of side VY opposite ∠XVY, we use the definition of sine: in a right triangle, the sine of an acute angle is equal to the length of the side opposite that angle divided by the length of the hypotenuse.

4. Substituting the known values, sin(67°) = VY / 34.

5. Looking up or calculating sin(67°) is approximately 0.9205, substituting into the equation, we get 0.9205 = VY / 34.

6. Solving the equation, we get VY = 0.9205 * 34 = 31.2971 mm.

8. Next, calculate the area of △WXY, given WX=28 mm.

9. According to the formula for the area of a triangle: Area = 1/2 * base * height, where the base is WX and the height is VY, thus Area = 1/2 * WX * VY.

10. Substituting the values, we get: Area = 1/2 * 28 mm * 31.3 mm.

11. Calculating, we get: Area = 438.2 square mm.

12. In summary, the length of VY is 31.3 mm, and the area of △WXY is 438.2 square mm.

Definition of Right Triangle

<note>In the diagram of this problem, in triangle VXY, angle ∠XVY is a right angle (90 degrees)</note>, therefore triangle VXY is a right triangle. <note>Side VX and side VY are the legs</note>, <note>side XY is the hypotenuse</note>.

Definition of Sine Function

In the figure of this problem, in the right triangle VXY, <note>angle ∠VXY is an acute angle</note>, <note>side VY is the opposite side of angle ∠VXY</note>, <note>side XY is the hypotenuse</note>. According to the definition of the sine function, <note>the sine value of angle ∠VXY is equal to the ratio of the opposite side VY to the hypotenuse XY</note>, that is, <note>sin(∠VXY) = VY / XY</note>.


Question:
What is the area of this figure?

 $\Box$  square miles

Answer: 50 square miles

1. Observing the figure, we can divide it into two rectangles, called Rectangle A and Rectangle B.

2. The base length of Rectangle A is 2 miles, and the height is 4 miles.

3. Using the rectangle area formula: Area = Base × Height, the area of Rectangle A is 2 miles × 4 miles = 8 square miles.

4. The base length of Rectangle B is 7 miles, and the height is 6 miles.

5. Using the rectangle area formula: Area = Base × Height, the area of Rectangle B is 7 miles × 6 miles = 42 square miles.

6. The total area of the figure is the sum of the areas of Rectangle A and Rectangle B: 8 square miles + 42 square miles = 50 square miles.

7. Through the above reasoning, the final answer is 50 square miles.

Definition of Rectangle

A quadrilateral is called a rectangle if and only if each of its interior angles is a right angle (90°) and its opposite sides are both parallel (∥) and congruent (≅).

In the figure of this problem, <note>Rectangle A and Rectangle B can both be considered rectangles</note>, satisfying the definition that <note>each pair of adjacent sides are perpendicular, and each interior angle is a right angle</note>. <note>Rectangle A has a base length of 2 miles and a height of 4 miles</note>; <note>Rectangle B has a base length of 7 miles and a height of 6 miles</note>.

Formula for the Area of a Rectangle

The area of a rectangle is equal to its length multiplied by its width.

<note>The length of the first rectangle is 2 miles</note>, <note>the width is 4 miles</note>, so the area = <note>2 miles × 4 miles = 8 square miles</note>. <note>The length of the second rectangle is 7 miles</note>, <note>the width is 6 miles</note>, so the area = <note>7 miles × 6 miles = 42 square miles</note>.


Question:
What is the area of this figure?

 $\Box$  square centimeters

Answer: 57 square centimeters

1. According to the given figure, use auxiliary lines to divide it into two rectangles. The boundary of the first rectangle A is the top and left sides, with a total width of 6 cm and a total height of 8 cm. The boundary of the second rectangle B is the right and bottom sides, with a width of 3 cm and a height of 3 cm.

2. Calculate the area of rectangle A. According to the formula for the area of a rectangle, the area is equal to the width multiplied by the height, so the area of rectangle A is 6 cm multiplied by 8 cm, which equals 48 square centimeters.

3. Determine the area of rectangle B. According to the same formula for the area of a rectangle, the area of rectangle B is 3 cm multiplied by 3 cm, which equals 9 square centimeters.

4. To find the total area of the entire figure, add the areas of rectangle A and rectangle B. That is, 48 square centimeters plus 9 square centimeters, resulting in a total area of 57 square centimeters.

5. Through the above reasoning, the final answer is 57 square centimeters.

Definition of Rectangle

<note>The top side length of rectangle A is 6 cm</note>, <note>the height is 8 cm</note>, <note>and it has four right angles</note>. <note>The bottom side length and height of rectangle B are both 3 cm</note>, <note>and it has four right angles</note>.

Formula for the Area of a Rectangle

In the figure of this problem, in rectangle A, <note>the sides of 6 cm and 8 cm are the length and width of the rectangle</note>, so <note>the area of the rectangle = 6 cm * 8 cm = 48 square cm</note>; in rectangle B, <note>the sides of 3 cm and 3 cm are the length and width of the rectangle</note>, so <note>the area of the rectangle = 3 cm * 3 cm = 9 square cm</note>.


Question:
What is the area of this figure?

 $\Box$  square kilometers

Answer: 38 square kilometers

1. Analyze the given figure, its shape is complex, so consider dividing it into easily calculable rectangular parts.

2. According to the information shown, the figure can be divided into two independent rectangles: Rectangle A and Rectangle B.

3. Observe Rectangle A, its width is 5 km and height is 4 km. According to the area formula for a rectangle Area = width × height, the area of Rectangle A is 5 km × 4 km = 20 km².

4. Next, observe Rectangle B, its width is 6 km and height is 3 km. According to the area formula for a rectangle, the area of Rectangle B is 6 km × 3 km = 18 km².

5. Add the areas of the two rectangles to obtain the area of the entire figure. Calculation: 20 km² + 18 km² = 38 km².

6. Through the above reasoning, the final answer is 38 km².

Definition of Rectangle

In the figure of this problem, rectangle A and rectangle B both belong to <note>矩形</note>. <note>Rectangle A has a width of 5 kilometers and a height of 4 kilometers</note>, while <note>rectangle B has a width of 6 kilometers and a height of 3 kilometers</note>. <note>The definition of a rectangle is a quadrilateral in which each interior angle is a right angle (90 degrees)</note> and the opposite sides are parallel and equal in length. <note>Rectangle A and rectangle B both have four interior right angles, and the opposite sides are parallel and equal in length respectively.</note> Therefore, we can divide the figure into these two rectangles for area calculation.

Formula for the Area of a Rectangle

<note>In rectangle A, the sides of 5 kilometers and 4 kilometers are the length and width of the rectangle</note>, so the area of the rectangle = 5 kilometers * 4 kilometers = 20 square kilometers. <note>In rectangle B, the sides of 6 kilometers and 3 kilometers are the length and width of the rectangle</note>, so the area of the rectangle = 6 kilometers * 3 kilometers = 18 square kilometers.


Question:
What is the area of this figure?

 $\Box$  square feet

Answer: 60 square feet

1. Analyze the figure and find that it can be divided into two rectangles: Rectangle A and Rectangle B. Based on the labeled side lengths, decompose the figure into Rectangle A and Rectangle B.

2. The width of Rectangle A is 8 ft, and the height is 3 ft. According to the area formula for a rectangle, the area of the rectangle is equal to the width multiplied by the height. Calculate the area of Rectangle A: 8×3＝24 square feet.

3. The width of Rectangle B is 4 ft, and the height is 9 ft. According to the area formula for a rectangle, calculate the area of Rectangle B: 4×9＝36 square feet.

4. Calculate the total area of the entire figure, which is the sum of the areas of the two rectangles. Add the area of Rectangle A and the area of Rectangle B: 24＋36＝60.

5. Through the above reasoning, the final answer is 60 square feet.

Definition of Rectangle

In the diagram of this problem, both Rectangle A and Rectangle B are rectangles. The side lengths of Rectangle A are 8 feet and 3 feet, and the side lengths of Rectangle B are 4 feet and 9 feet. <note>All interior angles of Rectangle A are right angles (90 degrees), and the 8-foot sides are parallel and equal in length, while the 3-foot sides are parallel and equal in length</note>. <note>All interior angles of Rectangle B are also right angles (90 degrees), and the 4-foot sides are parallel and equal in length, while the 9-foot sides are parallel and equal in length</note>.

Formula for the Area of a Rectangle

The width of <note>矩形A</note> is 8 feet, and the height is 3 feet, so <note>矩形A的面积 = 8英尺 × 3英尺 = 24平方英尺</note>. The width of <note>矩形B</note> is 4 feet, and the height is 9 feet, so <note>矩形B的面积 = 4英尺 × 9英尺 = 36平方英尺</note>.


Question:
Find the area of △UVW.

Write your answer as an integer or as a decimal rounded to the nearest tenth.
 $\Box$  km ^ 2

Answer: 313.7 km ^ 2

1. Given the angle measurements in triangle UVW: ∠ V = 21° and ∠ U = 28°, let VW be u, UW be v, and u = 33 km.

2. By the polygon interior angle sum theorem, the sum of the interior angles of a triangle is 180°, so ∠ V + ∠ U + ∠ W = 180°.

3. Substitute the given angle values: 21° + 28° + ∠ W = 180°, we can find ∠ W = 180° - 49° = 131°.

4. Now use the sine rule to solve for side length v. According to the sine rule, u/sin(U) = v/sin(V), thus 33/sin(28°) = v/sin(21°).

5. Calculate the sine values: sin(28°) ≈ 0.4695 and sin(21°) ≈ 0.3584.

6. Substitute into the calculation: 33/0.4695 ≈ v/0.3584, thus v ≈ 33/0.4695 * 0.3584 ≈ 25.1903 km.

7. Now that we have side length v and side length u as well as the included angle ∠ W, we can use the triangle area formula (using the sine function) to solve for the area: Area = 1/2 * u * v * sin(W).

8. Calculate the sine value: sin(131°) ≈ 0.7547.

9. Substitute into the area formula: Area = 0.5 * 33 * 25.1903 * 0.7547 ≈ 313.6877 km².

10. Round the area result to the nearest tenth: 313.7.

11. Through the above reasoning, the final answer is 313.7 square kilometers.

Definition of Triangle

In the figure of this problem, <note>triangle UVW is a geometric figure composed of three non-collinear points U, V, W and their connecting line segments UV, VW, WU</note>. <note>Points U, V, W are the three vertices of the triangle</note>, <note>line segments UV, VW, WU are the three sides of the triangle</note>.

Measurement of Angle

Align the baseline of the protractor with one side of the angle, then extend the other side to the protractor's scale. Determine the angle by reading the difference between the graduations of the two sides.

<note>Angle m∠V = 21°</note>, <note>Angle m∠U = 28°</note>, it can be determined that <note>Angle m∠W = 180°-21°-28°=131°</note>.

Polygon Interior Angle Sum Theorem

The sum of the interior angles of a polygon is equal to (n - 2) * 180°, where n represents the number of sides of the polygon.

In <note>triangle UVW</note>, <note>UVW is a polygon with 3 sides</note>, where <note>n represents the number of sides of the polygon</note>. According to the Polygon Interior Angle Sum Theorem, the sum of the interior angles of this polygon is equal to <note>(3-2) × 180°</note>. By knowing the degrees of two interior angles, the third interior angle can be calculated: <note>m∠W</note> = <note>180° - (m∠U + m∠V) = 180° - 49° = 131°</note>.

Sine Theorem

In <note>any triangle UVW</note>, <note>sides UV, VW, and WU</note> correspond to <note>angles ∠W, ∠U, and ∠V</note> respectively. According to the <note>Sine Theorem</note>, the ratio of the lengths of the sides to the sine of their opposite angles is equal and is equal to the <note>diameter of the circumscribed circle</note>, that is: <note>UV/sin(∠W)=VW/sin(∠U)=WU/sin(∠V) = 2r = D</note> (where <note>r is the radius of the circumscribed circle</note>, <note>D is the diameter</note>). In this problem, <note>side VW</note> can be determined using the <note>Sine Theorem</note>. Given <note>side UV and the sine value of its opposite angle</note>: <note>UV/sin(∠W) = VW/sin(∠U)</note>, that is <note>33/sin(28°) = VW/sin(21°)</note>. Then, by calculating the sine values, <note>VW ≈ 25.1903 km</note> can be obtained.

In the figure of this problem, in triangle UVW, <note>side UV and side VW are u and v respectively</note>, <note>angle ∠UVW is the included angle W between these two sides</note>. According to the triangle area formula, the area S of triangle UVW can be expressed as <note>S = (1/2) * u * v * sin(W)</note>, that is, S = (1/2) * <note>33</note> * <note>25.1903</note> * <note>sin(131°)</note> ≈ 313.6877 km².