Let's walk through this step-by-step:
1. Understanding the relationship of cosine with triangle dilation: The cosine of an angle in a triangle doesn't depend on the size of the triangle. This is a key property of trigonometric functions—they only depend on the angle measure, not on the side lengths directly.
2. Cosine of angle x in the original triangle: Given that in the original triangle, the cosine of angle [tex]\( x \)[/tex] is provided as [tex]\( \cos(x) = \frac{8}{17} \)[/tex].
3. Cosine of angle x in the dilated triangle: When the triangle is dilated (i.e., every side is scaled by the same factor, in this case by 2 times), the angle measures within the triangle remain unchanged. Consequently, the trigonometric ratios (sine, cosine, tangent, etc.) for these angles will also remain the same.
So, the dilation doesn't affect the cosine value of angle [tex]\( x \)[/tex]. It will remain the same. Therefore, the value of [tex]\( \cos(x) \)[/tex] in the dilated triangle is also [tex]\( \frac{8}{17} \)[/tex].
Thus, the answer is:
Answer for Blank 1: [tex]\(\frac{8}{17}\)[/tex]
