Question

question 1 (essay worth 10 points)
(05.02 mc)
triangle xyz was dilated by a scale factor of 2 to create triangle acb. xy = 2.5, and yz = 5.
part a: find tan ∠x. explain how tan ∠x compares to tan ∠a. (5 points)
part b: find ac and cb. you must show all work and calculations to receive full credit. (5 points)

Explanation:

Part A:

Step1: Recall the property of similar triangles

Since triangle \(XYZ\) is dilated by a scale factor of \(2\) to create triangle \(ACB\), \(\triangle XYZ\sim\triangle ACB\) (by the definition of dilation, which preserves angle - measures). For two similar triangles, corresponding angles are equal. So \(\angle X=\angle A\).

Step2: Recall the tangent function formula

The tangent of an angle in a right - triangle is defined as \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). For \(\angle X\) in \(\triangle XYZ\) and \(\angle A\) in \(\triangle ACB\), since \(\angle X = \angle A\), \(\tan\angle X=\tan\angle A\) (because the ratio of the opposite side to the adjacent side for equal angles in right - triangles is the same).

Part B:

Step1: Use the scale factor to find side lengths

Given the scale factor \(k = 2\) (from \(\triangle XYZ\) to \(\triangle ACB\)). In \(\triangle XYZ\), \(XY = 2.5\) and \(YZ=5\).
Since \(\frac{AC}{XY}=2\) (scale factor formula for corresponding sides of similar triangles \( \frac{\text{side in}\triangle ACB}{\text{side in}\triangle XYZ}=2\)), then \(AC = 2\times XY\). Substituting \(XY = 2.5\), we get \(AC=2\times2.5 = 5\).

Step2: Use the scale factor for the other side

Also, since \(\frac{CB}{YZ}=2\), substituting \(YZ = 5\), we get \(CB=2\times YZ\). So \(CB = 2\times5=10\).

Answer:

• Part A: \(\tan\angle X=\tan\angle A\) because \(\triangle XYZ\sim\triangle ACB\) (by dilation) and for similar right - triangles, the tangent of corresponding acute angles (since \(\angle X\) and \(\angle A\) are corresponding angles) is equal as \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\) and the ratios of corresponding sides (used in the tangent formula) are equal due to similarity.
• Part B: \(AC = 5\) and \(CB = 10\)