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[tex]$\triangle XYZ$[/tex] was reflected over a vertical line, then dilated by a scale factor of [tex]$\frac{1}{2}$[/tex], resulting in [tex]$\triangle X^{\prime}Y^{\prime}Z^{\prime}$[/tex]. Which must be true of the two triangles? Select three options.

A. [tex]$\triangle XYZ \sim \triangle X^{\prime}Y^{\prime}Z^{\prime}$[/tex]
B. [tex]$\angle XZY \approx \angle Y^{\prime}Z^{\prime}X^{\prime}$[/tex]
C. [tex]$\overline{YX} \approx \overline{Y^{\prime}X^{\prime}}$[/tex]
D. [tex]$XZ = 2X^{\prime}Z^{\prime}$[/tex]
E. [tex]$m\angle YXZ = 2m\angle Y^{\prime}X^{\prime}Z^{\prime}$[/tex]

Sagot :

GinnyAnsweridn
To determine which statements are true about the triangles, let's go through each option step-by-step.

### Option 1: [tex]$\triangle XYZ \sim \triangle X'Y'Z'$[/tex]

A similarity transformation consists of a sequence of rigid motions followed by a dilation. Reflecting and then dilating a triangle does maintain similarity since the shape is preserved even though the size changes. Therefore, [tex]$\triangle XYZ$[/tex] is similar to [tex]$\triangle X'Y'Z'$[/tex].

Verdict: True

### Option 2: [tex]$\angle XZY \approx \angle Y'Z'X'$[/tex]

When a triangle is reflected, the angles do not change; they remain congruent. This property holds irrespective of whether a dilation follows because dilations also do not affect angle measures.

Verdict: True

### Option 3: [tex]$\overline{YX} \approx \overline{Y'X'}$[/tex]

Reflection alone would keep the corresponding side lengths equal, but a subsequent dilation by a scale factor of [tex]$\frac{1}{2}$[/tex] changes the length of all sides in [tex]$\triangle X'Y'Z'$[/tex] to be half the length of the corresponding sides in [tex]$\triangle XYZ$[/tex]. Hence, the lengths are not approximately equal.

Verdict: False

### Option 4: [tex]$XZ = 2 X'Z'$[/tex]

Due to the dilation by a scale factor of [tex]$\frac{1}{2}$[/tex], every side in [tex]$\triangle X'Y'Z'$[/tex], including [tex]$X'Z'$[/tex], will be half the length of the corresponding side in [tex]$\triangle XYZ$[/tex]. Therefore, [tex]$XZ = 2X'Z'$[/tex].

Verdict: True

### Option 5: [tex]$m \angle YXZ = 2 m \angle Y'X'Z'$[/tex]

Reflecting or dilating a triangle does not change the measure of the angles. Since dilations preserve angle measures, the angle measures in [tex]$\triangle XYZ$[/tex] should match those in [tex]$\triangle X'Y'Z'$[/tex] and not be multiplied by a factor of 2.

Verdict: False

In conclusion, the true statements are:
- [tex]$\triangle XYZ \sim \triangle X'Y'Z'$[/tex]
- [tex]$\angle XZY \approx \angle Y'Z'X'$[/tex]
- [tex]$XZ = 2X'Z'$[/tex]
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