Question

△xyz was reflected over a vertical line, then dilated by a scale factor of 1/2, resulting in △xyz. which must be true of the two triangles? select three options. □ △xyz ~ △xyz □ ∠xzy ≅ ∠yzx □ overline{yx} ≅ overline{yx} □ xz = 2xz □ m∠yxz = 2m∠yxz

Explanation:

Step1: Similarity of triangles

Since $\triangle XYZ$ is reflected (a rigid transformation, which preserves shape and size, and thus angles) and then dilated (a similarity transformation). By the definition of similarity, if a figure is transformed by a sequence of rigid - transformations (reflections, rotations, translations) and dilations, the original figure ($\triangle XYZ$) and the transformed figure ($\triangle X'Y'Z'$) are similar. So, $\triangle XYZ\sim\triangle X'Y'Z'$.

Step2: Congruence of angles in similar triangles

In similar triangles, corresponding angles are congruent. For $\triangle XYZ\sim\triangle X'Y'Z'$, $\angle XZY$ and $\angle Y'Z'X'$ are corresponding angles. So, $\angle XZY\cong\angle Y'Z'X'$.

Step3: Relationship between side lengths

The scale factor of the dilation is $k=\frac{1}{2}$. If we consider the side lengths, the formula for the relationship between the side lengths of the original figure ($s$) and the dilated figure ($s'$) is $s'=ks$. Let $s = XZ$ and $s'=X'Z'$. Then $X'Z'=\frac{1}{2}XZ$, which can be rewritten as $XZ = 2X'Z'$.

For the option $\overline{YX}\cong\overline{Y'X'}$: Since there is a dilation with scale factor $\frac{1}{2}$, the lengths of the sides of $\triangle XYZ$ and $\triangle X'Y'Z'$ are not equal (except when the scale factor $k = 1$). For the option $m\angle YXZ=2m\angle Y'X'Z'$: In similar triangles, corresponding angles are equal, not in a multiple - of - two relationship. So, $m\angle YXZ=m\angle Y'X'Z'$.

Answer:

• $\triangle XYZ\sim\triangle X'Y'Z'$
• $\angle XZY\cong\angle Y'Z'X'$
• $XZ = 2X'Z'$