Question

two transformations are performed on δx in the coordinate plane. first, δx is rotated 90° counterclockwise around the origin to form image δx. then δx is dilated using a scale factor of \\(\frac{3}{4}\\) with the center of dilation at the origin to form image δx. which statement about δx and δx is true? a δx is similar to δx. b δx is congruent to δx. c δx is both congruent and similar to δx. d δx is neither congruent nor similar to δx.

Explanation:

1. Recall the properties of transformations:

• A rotation is a rigid transformation, which means it preserves the shape and size of the figure. So, \(\triangle X\) and \(\triangle X'\) are congruent (and thus also similar, since congruent figures are a special case of similar figures with a scale factor of 1).
• A dilation with a scale factor \(k\) (where \(k

eq1\)) changes the size of the figure but preserves the shape. So, \(\triangle X'\) and \(\triangle X''\) are similar (because dilation is a similarity transformation).

• By the transitive property of similarity, if \(\triangle X\sim\triangle X'\) and \(\triangle X'\sim\triangle X''\), then \(\triangle X\sim\triangle X''\).
• However, the dilation has a scale factor of \(\frac{3}{4}\), which is not equal to 1, so \(\triangle X\) and \(\triangle X''\) are not congruent (since congruent figures must have the same size, i.e., scale factor 1 for the transformation between them).

1. Analyze each option:

• Option A: Since rotation preserves similarity (and congruence) and dilation preserves similarity, \(\triangle X\) is similar to \(\triangle X''\). This is correct.
• Option B: Congruence requires the same size, but dilation with scale factor \(\frac{3}{4}\) changes the size, so \(\triangle X\) and \(\triangle X''\) are not congruent. Eliminate B.
• Option C: They are not congruent (due to dilation), so this is incorrect.
• Option D: They are similar (as explained), so this is incorrect.

Answer:

A. \(\triangle X\) is similar to \(\triangle X''\).