Certainly! To solve the problem, let's first understand the key concepts involved.
When a triangle undergoes dilation, i.e., it's scaled to be larger or smaller, the lengths of the sides of the triangle change. However, the angles of the triangle remain the same because dilation is a similarity transformation.
One important property in mathematics is that the cosine of an angle in a triangle depends solely on the size of the angle itself, not on the size of the triangle. That is, since cosine is a ratio of the sides of the triangle, even if the triangle is scaled (dilated), this ratio remains the same.
Given that:
- The cosine of angle x in the original triangle is [tex]\(\frac{8}{17}\)[/tex].
- The triangle is dilated to be two times as big as the original triangle.
We need to determine the value of the cosine of angle x in the dilated triangle. Here’s the detailed solution:
1. Understand that dilation affects the side lengths but does not affect the angle magnitudes. Therefore, the cosine ratio, which depends only on the angles, remains unchanged.
2. The original cosine value of angle x is given as [tex]\(\frac{8}{17}\)[/tex].
3. Because the dilation (scaling of the triangle) does not alter the cosine value of an angle, the cosine of angle x will remain the same in the dilated triangle.
Thus, the value of the cosine of angle x for the dilated triangle is:
[tex]\[
0.47058823529411764
\][/tex]
