Question:
In the rectangular prism shown below, which lines are parallel? Select all that apply.

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| $\overleftrightarrow{PQ}$ | and | $\overleftrightarrow{LM}$ |
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| $\overleftrightarrow{LM}$ | and | $\overleftrightarrow{MN}$ |
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| $\overleftrightarrow{PS}$ | and | $\overleftrightarrow{LP}$ |
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| $\overleftrightarrow{NR}$ | and | $\overleftrightarrow{MN}$ |

Answer:
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| \$\overleftrightarrow{PQ}\$ | and | \$\overleftrightarrow{LM}\$ |

1. First, observe the cuboid structure. According to the definition of a cuboid, the opposite sides on the same face are parallel.

2. Line segments \\(\overleftrightarrow{PQ}\\) and \\(\overleftrightarrow{LM}\\) are on face PLMQ. According to the definition of a rectangle, these two line segments are parallel because they are opposite sides of face PLMQ and do not intersect within the same plane.

3. Line segments \\(\overleftrightarrow{LM}\\) and \\(\overleftrightarrow{MN}\\) intersect at point M. According to plane geometry, if two lines in the same plane have a common point, they are not parallel.

4. Line segments \\(\overleftrightarrow{PS}\\) and \\(\overleftrightarrow{LP}\\) are also not parallel because they intersect at point P.

5. Line segments \\(\overleftrightarrow{NR}\\) and \\(\overleftrightarrow{MN}\\) are not parallel because they intersect at point N.

6. From the above reasoning, only \\(\overleftrightarrow{PQ}\\) and \\(\overleftrightarrow{LM}\\) are parallel lines.

7. After the above reasoning, the final answer is \\(\overleftrightarrow{PQ}\\) and \\(\overleftrightarrow{LM}\\).

Definition of Rectangular Prism

A rectangular prism is defined as a three-dimensional geometric figure formed by six rectangular faces, where three mutually adjacent faces respectively define the length, width, and height.

<note>A rectangular prism is enclosed by six rectangular faces</note>.<note>Three adjacent faces respectively define the length SR, width RQ, and height NR</note>. Each rectangular face is perpendicular to the others, and the areas of opposite faces are equal. The formula for calculating the volume of the rectangular prism is: <note>Volume = length SR × width RQ × height NR</note>.

Definition of Parallel Lines

Two lines in the same plane that do not intersect are called parallel lines.

<note>Line PQ and line LM</note> are located in the same plane, and they do not intersect, so according to the <note>definition of parallel lines</note>, <note>line PQ and line LM</note> are parallel lines.

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 quadrilateral <note>LMQP</note>, each interior angle <note>P, Q, L, and M</note> is a <note>right angle (90 degrees)</note>, and side <note>PQ is parallel and equal in length to side LM</note>, and side <note>PQ is parallel and equal in length to side LM</note>. Therefore, <note>the quadrilateral LMQP is a rectangle</note>.


Question:
The triangle V'W'X' is a dilation of the triangle VWX. What is the scale factor of the dilation?

Simplify your answer and write it as a proper fraction, an improper fraction, or a whole number.

 $\Box$

Answer: 3

1. First, determine the coordinates of the vertices of the original triangle. The coordinates of the vertices of triangle VWX are V(-2, 1), W(3, -2), and X(-2, -3).

2. Then, determine the coordinates of the vertices of the triangle V'W'X' obtained after the similarity transformation. The coordinates of the vertices of triangle V'W'X' are V'(-6, 3), W'(9, -6), and X'(-6, -9).

3. Choose a corresponding side to calculate. We choose sides VX and V'X'.

4. Calculate the length of side VX of the original triangle. The coordinates of points V and X are (-2, 1) and (-2, -3) respectively, so the length of VX is |1 - (-3)| = 4.

5. Calculate the length of the corresponding side V'X' of the triangle after the similarity transformation. The coordinates of points V' and X' are (-6, 3) and (-6, -9) respectively, so the length of V'X' is |3 - (-9)| = 12.

6. Calculate the scale factor of the similarity transformation, which is the ratio of the length of the new triangle's side to the length of the original triangle's side. The scale factor f = V'X'/VX = 12/4.

7. Simplify the obtained ratio 12/4 = 3.

8. Through the above reasoning, the final answer is 3.

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>In the figure of this problem, triangle VWX is a geometric figure composed of three non-collinear points V(-2, 1), W(3, -2), and X(-2, -3) and their connecting line segments VW, WX, VX.</note> Points <note>V, W, X are the three vertices of the triangle</note>, <note>and line segments VW, WX, VX are the three sides of the triangle</note>. After a similarity transformation, <note>triangle V'W'X' is a geometric figure composed of three non-collinear points V'(-6, 3), W'(9, -6), and X'(-6, -9) and their connecting line segments V'W', W'X', V'X'</note>. Points <note>V', W', X' are the three vertices of the triangle</note>, <note>and line segments V'W', W'X', V'X' are the three sides of the triangle</note>.

Coordinates of a Point

In a Cartesian coordinate system, the position of a point is represented by an ordered pair (x, y), where x denotes the horizontal (x-coordinate) and y denotes the vertical (y-coordinate).

<note>The coordinates of point V are (-2, 1)</note>, <note>The coordinates of point W are (3, -2)</note>, <note>The coordinates of point X are (-2, -3)</note>; <note>The coordinates of point V' are (-6, 3)</note>, <note>The coordinates of point W' are (9, -6)</note>, <note>The coordinates of point X' are (-6, -9)</note>.

Distance Formula Between Two Points

In a Cartesian coordinate plane, the distance \( d \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) can be calculated using the formula \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).

In this problem, the <note>length of segment VX</note> is calculated as |1 - (-3)| = 4, and the <note>length of segment V'X'</note> is calculated as |3 - (-9)| = 12.

Definition of Scale Factor in Rectangular Coordinate Systems

In a rectangular coordinate system, if a polygon xxx is transformed to xxx by scaling, then the scale factor is defined as the ratio of the length of any side of the scaled polygon to the length of the corresponding side of the original polygon.

<note>After scaling polygon VWX, we get polygon V'W'X'</note>, scale factor = <note>length of V'X' / length of VX</note>, which means scale factor = 3.


Question:
In the graph below, square L'M'N'O' is the image of LMNO after a dilation.

What are the scale factor and center of the dilation?
Simplify your answers and write them as fractions or whole numbers.

scale factor:  $\Box$

center of the dilation: ( $\Box$ ,  $\Box$ )

Answer:
scale factor: 3
center of the dilation: (-6, 9)

1. To determine the scale factor of enlargement, first calculate the length of segment LM. According to the coordinate point formula, the endpoints of LM are L (-6,6) and M (-3,7). Substitute into the distance formula: LM = √((−3−(−6))^2+(7−6)^2) = √(3^2+1^2) = √(9+1) = √10.

2. Then calculate the length of segment L'M'. The endpoints of L'M' are L' (-6,0) and M' (3,3). Substitute into the distance formula: L'M' = √((3−(−6))^2+(3−0)^2) = √(9^2+3^2) = √(81+9) = √90 = 3√10.

3. To determine the scale factor, calculate the ratio of the lengths of L'M' to LM, obtaining the scale factor: L'M'/LM = (3√10)/√10 = 3.

4. Determine the center of enlargement. According to the properties of enlargement, each point in the figure will expand along the line passing through the original point and the enlarged point. Take the line equation of L and L'. Using the slope formula, calculate the slope of the line m = (0-6)/(-6+6) = ∞, and through the X coordinate of vertex L', it is clear that it is a vertical line passing through X=-6.

5. Similarly, determine the line equation of points M and M'. The slope of the line m = (3-7)/(3+3) = (-4)/6 = -2/3, and the linear equation is y-7 = -2/3(x+3).

6. Solve the intersection of the above two lines. Substitute X=-6 into y-7 = -2/3 * (x+3), obtaining y = -2/3 * (-6+3)+7 = 9.

7. By proving the intersection point, the center of enlargement is (-6,9).

8. Through the above reasoning, the final answer is that the scale factor is 3, and the center of enlargement is (-6,9).

In this problem diagram, <note>the distance formula for segment LM</note> is: LM = √((-3 - (-6))^2 + (7 - 6)^2) = √(3^2 + 1^2) = √10. <note>The distance formula for segment L'M'</note> is: L'M' = √((3 - (-6))^2 + (3 - 0)^2) = √(9^2 + 3^2) = √90 = 3√10.

In the problem diagram, the polygon LMNO is scaled to obtain the polygon LM'N'O', the scale factor = <note>length of L'M' / length of LM</note>, i.e., the scale factor = 3.