Question: Given the circle below with secant XW and tangent UV, find the length of XW. Round to the nearest…

Given the circle below with secant XW and tangent UV, find the length of XW. Round to the nearest tenth if necessary.

Solution

To solve for the length of \( \overline{XW} \), we can use the Power of a Point theorem, which states that for a point outside a circle, the product of the whole secant segment and its external part is equal to the square of the tangent segment. In this case: \[ \begin{cases} \overline{XW} \cdot \overline{WV} = \overline{UV}^2 & \end{cases} \] Given: \[ \overline{WV} = 6, \quad \overline{UV} = 13 \] Substitute the known values into the equation: \[ \overline{XW} \cdot 6 = 13^2 \] Calculate \( 13^2 \): \[ 13^2 = 169 \] Now, substitute back: \[ \overline{XW} \cdot 6 = 169 \] Solve for \( \overline{XW} \): \[ \overline{XW} = \frac{169}{6} \] Calculate the division: \[ \overline{XW} \approx 28.1667 \] Round to the nearest tenth: \[ \overline{XW} \approx 28.2 \] Therefore, the length of \( \overline{XW} \) is approximately 28.2 units.