Question

quadrilateral abcd is dilated with center p and a scale factor of 3 to create quadrilateral abcd. select all of the statements that must be true. a the length of side cd is 3 times the length of side cd. b the measure of angle bca is 3 times the measure of angle bca. c the length of side bc is 3 times the length of side bc. d the measure of angle cda is equal to the measure of angle cda.

Explanation:

Step1: Understand the properties of dilation

Dilation is a transformation that changes the size of a figure but not its shape. The scale factor \(k = 3\). If a figure \(ABCD\) is dilated to \(A'B'C'D'\) with scale factor \(k\), then the length of a side \(s'\) of the dilated figure is related to the length of the corresponding side \(s\) of the original figure by the formula \(s'=k\times s\). Also, the measures of angles in the original figure and the dilated figure are equal because dilation preserves angle - measures.

Step2: Analyze each option

• Option A:

If \(k = 3\), then \(C'D'=3\times CD\), not \(CD = 3\times C'D'\). So, this option is false.

• Option B:

Since dilation preserves angle - measures, \(\angle B'C'A'=\angle BCA\), not \(\angle B'C'A' = 3\times\angle BCA\). So, this option is false.

• Option C:

Using the formula \(s'=k\times s\) with \(k = 3\) and \(s = BC\), \(s'=B'C'\), we get \(B'C'=3\times BC\). So, this option is true.

• Option D:

Because dilation preserves angle - measures, \(\angle C'D'A'=\angle CDA\). So, this option is true.

Answer:

C. The length of side \(B'C'\) is 3 times the length of side \(BC\); D. The measure of angle \(C'D'A'\) is equal to the measure of angle \(CDA\)