To solve this problem, we need to analyze the relationship between triangle XYZ and triangle ACB after the dilation, and also how to calculate the measures of the segments CB and AB.
Part A: Relationship Between Trigonometric Ratios
Since triangle ACB is a dilation of triangle XYZ with a scale factor of 2, all corresponding side lengths in triangle ACB are twice the lengths of the corresponding sides in triangle XYZ. This means that the angles remain congruent and the ratios of the lengths of the sides are preserved despite the dilation.
Given that \(\sin \angle X = \frac{5}{5.59}\), we can calculate the other trigonometric ratios for triangle XYZ since angles A and X are congruent (and thus have the same sine, cosine, and tangent ratios).
Let's denote the corresponding side opposite angle X in triangle XYZ as \(a\), the side opposite angle Y (which is a right angle) as \(b\), and the hypotenuse (side opposite the right angle) as \(c\).
Using the sine function: \[ \sin X = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{a}{c} \] From this: \[ a = c \cdot \sin X \]
For the dilated triangle ACB, the sides are double the measures of the corresponding sides in triangle XYZ. Therefore: \[ CB = 2a \] and \[ AB = 2c \]
The sine of angle ACB will also maintain the same ratio because: \[ \sin ACB = \frac{CB}{AB} \]
Thus, both the original triangle XYZ and the dilated triangle ACB maintain the same trigonometric ratios because they share angles and scaled measures maintain the ratio of side lengths.
Part B: Finding Measures of Segments CB and AB
To find the measures of segments CB and AB, we need to know either the lengths of sides of triangle XYZ or any related information about it. Let's denote the lengths of sides of triangle XYZ as follows:
- Let \(h\) be the hypotenuse (length corresponding to side \(c\))-> can be calculated from the \(\sin X\).
Using \(\sin \angle X = \frac{5}{5.59}\): \[ h = \frac{a}{\sin X} \]
To perform the calculation:
-
Measure \(a\): Let's calculate \(a\):
- If \(a = 5\) (for simplicity), we can find the length of side \(c\):
\[ c = \frac{a}{\sin X} = \frac{5}{\frac{5}{5.59}} = 5.59 \]
Using this \(c\), now we can find \(CB\) and \(AB\):
- \(CB = 2a = 2 \cdot 5 = 10\)
- \(AB = 2c = 2 \cdot 5.59 ≈ 11.18\)
Thus, the measures of segments CB and AB are:
- CB = 10
- AB ≈ 11.18
By clearly showing the calculations and relationships, we have demonstrated how the properties of trigonometric ratios remain consistent between the similar triangles and how to compute segment lengths based on those properties. If more specifics on triangle XYZ are given, more accurate lengths can be derived.