Question

Help. use the figure shown to the right to find the value of x

Answer 1

`\begin{aligned}x &= 16\sqrt3 \\ &\approx 27.7\end{aligned}`

Explanation:

We can see that the longer leg (a) of a right triangle is half of the circle's radius. Since we are given the other two sides of the triangle (shorter leg and hypotenuse), we can solve for the length of the longer leg using the Pythagorean Theorem:

`a^2 + b^2 = c^2`

↓ plugging in the given values

`a^2 + 2^2 = 14^2`

↓ subtracting 2² from both sides

`a^2 = 14^2 - 2^2`

`a^2 = 196 - 4`

`a^2 = 192`

↓ taking the square root of both sides

`a = \sqrt{192`

↓ simplifying the square root

`a = \sqrt{2^6 \cdot 3`

`a = 2^(\, 6 / 2) \cdot \sqrt3`

`a = 2^3\sqrt3`

`a = 8\sqrt3`

Now, we can solve for the radius (x) using the fact that the longer leg of the triangle is half of it.

`a = (1)/(2)x`

↓ plugging in the a-value we solved for

`8\sqrt3 = \frac{1}2x`

↓ multiplying both sides by 2

`\boxed{x = 16\sqrt3}`