Question

question 1 - 29
rectangle pqrs is shown below. point c is the center of the rectangle.
maggie claims that there are transformations that preserve the length of the rectangles sides. which of the following transformations would not be used to support maggies claim?

• a reflection over the side (overline{rs})
• a rotation of (90^{circ}) clockwise about vertex (q)
• a dilation of scale factor 1 through center (c)
• a vertical stretch of scale factor 2 through center (c)

Explanation:

Step1: Analyze reflection

A reflection is a rigid transformation. Rigid transformations preserve the lengths of the sides of a figure. So, a reflection over the side \( \overline{RS} \) will preserve the side - lengths of rectangle \( PQRS \).

Step2: Analyze rotation

A rotation is a rigid transformation. A rotation of \( 90^{\circ} \) clockwise about vertex \( Q \) will preserve the side - lengths of rectangle \( PQRS \) since rigid transformations (rotations, reflections, translations) keep the shape and size (side - lengths) of a figure unchanged.

Step3: Analyze dilation with scale factor 1

A dilation with a scale factor \( k = 1 \) is the identity transformation. If we have a dilation of scale factor \( 1 \) through center \( C \), the image of the rectangle \( PQRS \) is congruent to the original rectangle. So, the side - lengths are preserved. The formula for dilation is \( (x,y)\to k(x - x_0)+x_0,k(y - y_0)+y_0 \), when \( k = 1 \), \( (x,y)\to(x,y) \).

Step4: Analyze vertical stretch

A vertical stretch is a non - rigid transformation. The formula for a vertical stretch about a center \( (x_0,y_0) \) with scale factor \( k \) is \( (x,y)\to(x,k(y - y_0)+y_0) \). When \( k = 2 \), the vertical side - lengths of the rectangle will change. For example, if a vertical side has endpoints \( (x_1,y_1) \) and \( (x_1,y_2) \) in the original rectangle, after a vertical stretch about a center \( C(x_0,y_0) \), the new \( y \) - coordinates will be \( y_1'=2(y_1 - y_0)+y_0=2y_1 - y_0 \) and \( y_2'=2(y_2 - y_0)+y_0=2y_2 - y_0 \), and \( |y_2' - y_1'|=2|y_2 - y_1| \).

Answer:

a vertical stretch of scale factor 2 through center \( C \)