Question

if △abc is a dilation of △xyz, select all statements that must be true. a. m∠a = m∠x b. m∠c = m∠z c. (overline{ac}congoverline{xz}) d. area of △abc is greater than area of △xyz e. perimeter of △abc is less than perimeter of △xyz

Explanation:

Step1: Recall dilation property

Dilation is a similarity - transformation. Similar triangles have equal corresponding angles.

Step2: Analyze angle - equality

Since $\triangle ABC$ is a dilation of $\triangle XYZ$, $\angle A$ and $\angle X$, $\angle C$ and $\angle Z$ are corresponding angles. So $m\angle A=m\angle X$ and $m\angle C = m\angle Z$.

Step3: Analyze side - length and perimeter/area

The lengths of corresponding sides of similar triangles are in proportion. We cannot say $\overline{AC}\cong\overline{XZ}$ without knowing the scale factor. Also, if the scale factor $k> 1$, the area and perimeter of $\triangle ABC$ are greater than those of $\triangle XYZ$, if $0 < k<1$, the area and perimeter of $\triangle ABC$ are less than those of $\triangle XYZ$.

Answer:

A. $m\angle A=m\angle X$
B. $m\angle C = m\angle Z$