Question

triangle pqr is rotated, translated, and dilated to obtain triangle pqr as shown. which statement about the two figures is correct? a. the two figures are similar because each angle measure of the image is \\( \frac { 2 } { 3 } \\) times the corresponding angle measure of the preimage and all the pairs of corresponding sides are congruent. b. the two figures are similar because each side length of the image is \\( \frac { 2 } { 3 } \\) times the corresponding side length of the preimage and all the pairs of corresponding angles are congruent. c. the two figures are similar because each angle measure of the image is \\( \frac { 3 } { 2 } \\) times the corresponding angle measure of the preimage and all the pairs of corresponding sides are congruent. d. the two figures are similar because each side length of the image is \\( \frac { 3 } { 2 } \\) times the corresponding side length of the preimage and all the pairs of corresponding angles are congruent.

Explanation:

Step1: Recall the properties of similar triangles

Similar triangles have congruent corresponding angles and proportional corresponding sides. Rotation and translation do not change the shape and size (except for dilation which changes the size proportionally).

Step2: Analyze the side - length ratio

Let's assume a side length of the pre - image (triangle \(PQR\)). If we count the grid units for a side (say \(PQ\) in the pre - image and \(P'Q'\) in the image). Suppose the length of a side of \(\triangle PQR\) is \(x\) and the length of the corresponding side of \(\triangle P'Q'R'\) is \(y\). By counting the grid units (for example, if \(PQ = 2\) units and \(P'Q'=3\) units), the ratio of the side length of the image to the pre - image is \(\frac{y}{x}=\frac{3}{2}\). Angles are preserved under rotation, translation, and dilation (since dilation is a similarity transformation). In a similarity transformation, if \(\triangle ABC\sim\triangle A'B'C'\), then \(\angle A=\angle A'\), \(\angle B = \angle B'\), \(\angle C=\angle C'\) and \(\frac{A'B'}{AB}=\frac{B'C'}{BC}=\frac{A'C'}{AC}\) (the scale factor).

Answer:

D. The two figures are similar because each side length of the image is \(\frac{3}{2}\) times the corresponding side length of the preimage and all the pairs of corresponding angles are congruent.