$\Delta XYZ$ was reflected over a vertical line, then dilated by a scale factor of $\frac{1}{2}$, resulting in $\Delta X^{\prime} Y^{\prime} Z^{\prime}$. Which must be true of the two triangles? Select three options.
$\Delta XYZ \sim \Delta X^{\prime} Y^{\prime} Z^{\prime}$
$\angle XZY \cong \angle Y^{\prime} Z^{\prime} X^{\prime}$
$\overline{YX} \cong \overline{Y^{\prime} X^{\prime}}$
$XZ=2 X^{\prime} Z^{\prime}$
$m \angle YXZ =2 m \angle Y ^{\prime} X ^{\prime} Z ^{\prime}$
Answer (1)
Reflection and dilation are transformations applied to Δ X Y Z to obtain Δ X ′ Y ′ Z ′ .
Similarity is preserved through both reflection and dilation, thus Δ X Y Z ∼ Δ X ′ Y ′ Z ′ .
Angle measures are preserved, but the order of vertices is reversed by reflection, so ∠ XZ Y ≅ ∠ Y ′ Z ′ X ′ .
Dilation scales side lengths, making XZ = 2 X ′ Z ′ .
The three true statements are: Δ X Y Z ∼ Δ X ′ Y ′ Z ′ , ∠ XZ Y ≅ ∠ Y ′ Z ′ X ′ , and XZ = 2 X ′ Z ′ .
Δ X Y Z ∼ Δ X ′ Y ′ Z ′ , ∠ XZ Y ≅ ∠ Y ′ Z ′ X ′ , XZ = 2 X ′ Z ′
Explanation
Analyze the transformations First, let's analyze the transformations applied to triangle X Y Z . A reflection over a vertical line preserves the shape and size of the triangle, but it reverses the orientation. A dilation by a scale factor of 2 1 changes the size of the triangle but preserves its shape.
Check similarity Since reflection and dilation both preserve the shape of the triangle, Δ X Y Z is similar to Δ X ′ Y ′ Z ′ . Therefore, the statement Δ X Y Z ∼ Δ X ′ Y ′ Z ′ is true.
Check angle congruence Reflection changes the order of vertices, and dilation preserves angles. So, if we consider the angle XZ Y in the original triangle, after reflection, the order of vertices will be reversed. Therefore, ∠ XZ Y ≅ ∠ Y ′ Z ′ X ′ is true.
Check side length congruence Since dilation changes side lengths, Y X is not congruent to Y ′ X ′ . Congruence implies that the lengths are the same, but the dilation changes the side lengths by a factor of 2 1 . Therefore, the statement Y X ≅ Y ′ X ′ is false.
Check side length relationship Dilation scales the side lengths by a factor of 2 1 . This means that XZ = 2 X ′ Z ′ . Therefore, the statement XZ = 2 X ′ Z ′ is true.
Check angle measure relationship Dilation preserves angle measures, and reflection also preserves angle measures. Therefore, m ∠ Y XZ = m ∠ Y ′ X ′ Z ′ . The statement m ∠ Y XZ = 2 m ∠ Y ′ X ′ Z ′ is false.
Final Answer Based on the analysis, the three true statements are:
Δ X Y Z ∼ Δ X ′ Y ′ Z ′
∠ XZ Y ≅ ∠ Y ′ Z ′ X ′
XZ = 2 X ′ Z ′
Examples
Understanding transformations like reflections and dilations is crucial in fields like computer graphics and architecture. For instance, when designing a building, architects use these transformations to create symmetrical designs (reflections) or to scale blueprints to different sizes (dilations) while maintaining the proportions of the original design.
