To determine the value of the tangent of angle [tex]\( x \)[/tex] after dilation, let's begin by understanding the properties of the tangent function and the concept of dilation.
1. Understanding Tangent Function:
- The tangent of an angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side. Mathematically, it is given as:
[tex]\[
\tan(x) = \frac{\text{opposite side}}{\text{adjacent side}}
\][/tex]
2. Initial Tangent Calculation:
- Given that [tex]\(\tan(x) = \frac{22}{5}\)[/tex], this ratio represents the relationship between the opposite and adjacent sides of the original triangle.
3. Effect of Dilation on Tangent:
- When a triangle is dilated, its sides are proportionately scaled up or down. In this case, the triangle is dilated to be twice its original size.
- For the tangent function, the ratio of the opposite side to the adjacent side remains the same, because both the opposite and adjacent sides are scaled by the same factor (in this case, 2).
- Therefore, the ratio [tex]\(\tan(x)\)[/tex] does not change with dilation.
4. Value of Tangent After Dilation:
- Since dilation does not affect the ratio of the sides, the value of [tex]\(\tan(x)\)[/tex] for the dilated triangle is the same as that for the original triangle.
Therefore, the value of [tex]\(\tan(x)\)[/tex] after the triangle is dilated is:
[tex]\[
\frac{22}{5}
\][/tex]
So, the answer for Blank 1 is:
[tex]\[
4.4
\][/tex]
