Question

practice problems

1. triangle abc is dilated using d as the center of dilation with scale factor 2.

the image is triangle abc.
clare says the 2 triangles are congruent, because their angle measures are the same. do you agree? explain your reasoning.

Explanation:

Step1: Recall the properties of dilation

Dilation is a transformation that changes the size of a figure but not its shape. When a figure is dilated with a scale factor \(k
eq1\), the lengths of its sides change. The formula for dilation is \(A'B' = k\times AB\), \(B'C'=k\times BC\), \(A'C' = k\times AC\). Here \(k = 2\).

Step2: Recall the definition of congruent triangles

Two triangles are congruent if all their corresponding sides and angles are equal. For triangle \(ABC\) and \(A'B'C'\), \(\angle A=\angle A'\), \(\angle B=\angle B'\), \(\angle C=\angle C'\) (dilation preserves angles). But \(A'B'=2AB\), \(B'C' = 2BC\), \(A'C'=2AC\).

Answer:

No, I do not agree. While dilation preserves the measures of angles (so \(\angle A=\angle A'\), \(\angle B=\angle B'\), \(\angle C=\angle C'\)), it changes the lengths of the sides. Since \(A'B' = 2AB\), \(B'C'=2BC\), \(A'C'=2AC\) (by the property of dilation with scale factor \(k = 2\)), the sides of \(\triangle ABC\) and \(\triangle A'B'C'\) are not equal. Congruent triangles require both equal - angle measures and equal - side lengths. So \(\triangle ABC\) and \(\triangle A'B'C'\) are not congruent.