To determine the value of the cosine of angle [tex]\( x \)[/tex] after the triangle has been dilated, it is important to understand some key geometric principles.
1. Cosine of an Angle and Dilations:
- The cosine of an angle in a triangle is defined as the ratio of the length of the adjacent side to the hypotenuse.
- When a triangle is dilated (enlarged or reduced) by a certain factor, the shapes of the triangles remain similar. This means that all corresponding angles remain the same.
- Since the angles do not change during dilation, the ratios of the lengths of corresponding sides also remain the same. Therefore, the cosine of the angle remains unchanged.
Given:
- Original value of [tex]\(\cos(x) = \frac{8}{17}\)[/tex].
Since the dilation does not affect the cosine of the angle:
The value of [tex]\(\cos(x)\)[/tex] for the dilated triangle remains:
[tex]\[
\cos(x) = \frac{8}{17}
\][/tex]
So, for the dilated triangle, the cosine of angle [tex]\( x \)[/tex] is [tex]\(\boxed{8/17}\)[/tex].
