Question

triangle abc is translated by the rule (x + 1, y + 1) and then dilated by a scale factor of 2 centered at the origin. which statement describes the properties of triangles abc and abc after the transformations? ∠c and ∠c are congruent after both the translation and the dilation. ∠c and ∠c are congruent after the dilation, but not after the translation. $overline{ac}$ and $overline{ac}$ are congruent after both the translation and the dilation. $overline{ac}$ and $overline{ac}$ are congruent after the dilation, but not after the translation.

Explanation:

Step1: Recall translation property

Translation is a rigid - motion. It preserves side - lengths and angle - measures. So, after translation, corresponding angles and sides of the original triangle and the translated triangle are congruent.

Step2: Recall dilation property

Dilation changes the side - lengths by a scale factor but preserves angle - measures. When a triangle is dilated, the angles of the original triangle and the dilated triangle are congruent.

Step3: Analyze angle congruence

Since translation and dilation both preserve angle - measures, ∠C and ∠C" are congruent after both the translation and the dilation.

Step4: Analyze side - length congruence

After translation, side - lengths are congruent. But after dilation with a scale factor of 2, the side - lengths of the dilated triangle are twice the side - lengths of the pre - dilated triangle. So, $\overline{AC}$ and $\overline{A''C''}$ are not congruent after the dilation.

Answer:

A. ∠C and ∠C" are congruent after both the translation and the dilation.