To solve this problem, we need to understand the effect of dilation on the cosine of an angle.
### Step-by-Step Solution:
1. Cosine Ratio Definition:
- The cosine of an angle in a right triangle is defined as the ratio of the length of the adjacent side to the hypotenuse. Mathematically, it is given by:
[tex]\[
\cos(x) = \frac{\text{adjacent}}{\text{hypotenuse}}
\][/tex]
2. Given Information:
- We are told that [tex]\(\cos(x)\)[/tex] is [tex]\(\frac{8}{17}\)[/tex]. This means that for a triangle with sides adjacent to and hypotenuse with lengths 8 and 17 respectively, the cosine of angle [tex]\(x\)[/tex] is:
[tex]\[
\cos(x) = \frac{8}{17}
\][/tex]
3. Effect of Dilation on Cosine:
- Dilation scales the lengths of the sides of the triangle by the same factor but does not change the value of the cosine of the angle. This is because the ratio of the adjacent side to the hypotenuse remains the same.
4. Conclusion:
- Hence, if the triangle is dilated to be two times as big, the lengths of the adjacent side and the hypotenuse will both be doubled to 16 and 34 respectively, but the ratio (cosine value) will not change.
So, the value of [tex]\(\cos(x)\)[/tex] for the dilated triangle remains:
[tex]\[
\frac{8}{17}
\][/tex]
Final Answer:
[tex]\(\frac{8}{17}\)[/tex]
