Question

question 8
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.
ac and ac are congruent after both the translation and the dilation.
ac and ac are congruent after the dilation, but not after the translation.

Explanation:

Step1: Understand translation property

Translation is a rigid - motion. In a translation $(x,y)\to(x + 1,y-1)$, the shape and size of the triangle are preserved. So, corresponding angles and sides of the original triangle $ABC$ and the translated triangle are congruent. That is, $\angle C$ and the corresponding angle in the translated triangle are congruent, and $\overline{AC}$ and the corresponding side in the translated triangle are congruent.

Step2: Understand dilation property

Dilation with a scale factor of $k = 2$ centered at the origin changes the size of the triangle. The side - lengths of the dilated triangle $A''B''C''$ are $k$ times the side - lengths of the pre - dilated triangle. So, $\overline{AC}$ and $\overline{A''C''}$ are not congruent after dilation. But angles are preserved under dilation. So, $\angle C$ and $\angle C''$ are congruent after both the translation and the dilation.

Answer:

$\angle C$ and $\angle C''$ are congruent after both the translation and the dilation.