Question

1. dilate abcd by scale factor of ½ and relabel the new points.
2. after the transformation. select all the statements that must be true.

a. $\frac{ad}{ad} = \frac{1}{2}$

b. $\frac{pc}{pc} = \frac{1}{2}$

c. angle b and b are congruent

d. angle d and d are supplementary

e. $\frac{ad}{ad} = \frac{1}{2}$

f. $\frac{pd}{pd} = \frac{pc}{pc}$

Explanation:

Step1: Recall Dilation Properties

Dilation is a similarity transformation. For a dilation with scale factor \( k = \frac{1}{2} \):

• Corresponding side lengths are in ratio \( k \), so \( \frac{\text{image side}}{\text{original side}} = k \).
• Distances from the center of dilation \( P \) to corresponding points are in ratio \( k \), so \( \frac{PC'}{PC} = \frac{PD'}{PD} = k \).
• Angles are preserved (congruent) in dilation.

Step2: Analyze Option a

\( A'D' \) is the image of \( AD \), so \( \frac{A'D'}{AD} = \frac{1}{2} \). This is true.

Step3: Analyze Option b

\( C' \) is the image of \( C \) under dilation from \( P \), so \( \frac{PC'}{PC} = \frac{1}{2} \). This is true.

Step4: Analyze Option c

Angles are congruent in dilation, so \( \angle B \cong \angle B' \). This is true.

Step5: Analyze Option d

Angles are congruent (not supplementary) in dilation, so \( \angle D \) and \( \angle D' \) are congruent, not supplementary. This is false.

Step6: Analyze Option e

\( \frac{AD}{A'D'} = \frac{1}{1/2} = 2 \), not \( \frac{1}{2} \). This is false.

Step7: Analyze Option f

\( \frac{PD'}{PD} = \frac{1}{2} \) and \( \frac{PC'}{PC} = \frac{1}{2} \), so \( \frac{PD'}{PD} = \frac{PC'}{PC} \). This is true.

Answer:

a. \(\boldsymbol{\frac{A'D'}{AD} = \frac{1}{2}}\), b. \(\boldsymbol{\frac{PC'}{PC} = \frac{1}{2}}\), c. Angle \( B \) and \( B' \) are congruent, f. \(\boldsymbol{\frac{PD'}{PD} = \frac{PC'}{PC}}\)