Question

kaitlyn drew $\triangle pqr$ on a coordinate grid. then she drew $\triangle pqr$, the result of a dilation of $\triangle pqr$ with a scale factor of $\frac{2}{3}$. how are $\triangle pqr$ and $\triangle pqr$ related?\
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a. they are similar but not congruent.\
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b. they are both similar and congruent.\
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c. they are neither similar nor congruent.\
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d. they are congruent but not similar

Explanation:

Step1: Recall dilation properties

Dilation is a transformation that scales a figure by a scale factor. For a scale factor \( k=\frac{2}{3}\) (not equal to 1), the image and pre - image have the same shape (so they are similar) but different sizes (so not congruent, since congruent figures have the same size and shape).

Step2: Analyze similarity and congruence

• Similarity: Dilation preserves the shape, so corresponding angles are equal and corresponding sides are in proportion (ratio equal to the scale factor). Thus, \(\triangle PQR\) and \(\triangle P'Q'R'\) are similar.
• Congruence: Congruent figures must have corresponding sides equal in length. Since the scale factor is \(\frac{2}{3}

eq1\), the sides of \(\triangle P'Q'R'\) are \(\frac{2}{3}\) the length of the sides of \(\triangle PQR\), so they are not congruent.

Answer:

A. They are similar but not congruent.