Question

triangle xyz was dilated by a scale factor of 2 to create triangle acb and \\(\cos \angle x = \frac{2.5}{5.59}\\).

part a: use complete sentences to explain the special relationship between the trigonometric ratios of triangles xyz and acb. you must show all work and calculations to receive full credit. (5 points)

part b: explain how to find the measures of segments ac and ab. you must show all work and calculations to receive full credit. (5 points)

Explanation:

Explain trigonometric ratios under dilation

Using the Right Triangle Trigonometry and Cosine Ratio knowledge points.
Dilation preserves angle measures, meaning \(\triangle XYZ \sim \triangle ACB\).
Corresponding angles are congruent, so \(\angle X \cong \angle A\).
Since the angles are equal, their trigonometric ratios are equal:

$$ \cos \angle A = \cos \angle X = \frac{2.5}{5.59} $$

Calculate side lengths of the original triangle

Using the Cosine Ratio knowledge point.
In right \(\triangle XYZ\) with right angle at \(Y\):

$$ \cos \angle X = \frac{XY}{XZ} = \frac{2.5}{5.59} $$

Thus, the side lengths of \(\triangle XYZ\) are proportional to:

$$ XY = 2.5,\quad XZ = 5.59 $$

Apply scale factor to find dilated lengths

Using the Right Triangle Trigonometry knowledge point.
The scale factor of dilation from \(\triangle XYZ\) to \(\triangle ACB\) is \(k = 2\).
Corresponding vertices map as \(X \to A\), \(Y \to C\), and \(Z \to B\).
Therefore, the corresponding side lengths are scaled by 2:

$$ AC = 2 \cdot XY = 2 \cdot 2.5 = 5 $$

$$ AB = 2 \cdot XZ = 2 \cdot 5.59 = 11.18 $$

Answer:

Part A

Triangle \(XYZ\) is dilated by a scale factor of 2 to create triangle \(ACB\). Because dilation preserves angle measures, the two triangles are similar (\(\triangle XYZ \sim \triangle ACB\)), meaning corresponding angles are congruent (\(\angle X \cong \angle A\)). Since trigonometric ratios depend only on the angle measure, the trigonometric ratios of corresponding angles in both triangles are equal. Therefore, \(\cos \angle A = \cos \angle X = \frac{2.5}{5.59}\).

Part B

To find the measures of segments \(AC\) and \(AB\), we use the definition of the cosine ratio in right triangle \(XYZ\), where \(\cos \angle X = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{XY}{XZ} = \frac{2.5}{5.59}\). This gives the side lengths of the original triangle as \(XY = 2.5\) and \(XZ = 5.59\). Since triangle \(XYZ\) is dilated by a scale factor of 2 to create triangle \(ACB\), we multiply the corresponding side lengths of triangle \(XYZ\) by 2:

$$ AC = 2 \cdot XY = 2 \cdot 2.5 = 5 $$

$$ AB = 2 \cdot XZ = 2 \cdot 5.59 = 11.18 $$