Question

100 Points! Geometry question. Photo attached. Please show as much work as possible. Thank you!

Answer 1

`x = (9)/(2)=4.5`

Perimeter = 52

Explanation:

A tangent is a straight line that touches a circle at only one point.

The given diagram shows a circle with three points of tangency: S, T and U.

According to the Two-Tangent Theorem, if two tangents to a circle meet at one exterior point, the tangent segments are congruent.

The exterior points are P, Q and R. Therefore, the congruent segments are:

`\overline{PS} = \overline{PT} = 9`

`\overline{QT} = \overline{QU} = 4`

`\overline{RS} = \overline{RU} = 13`

To find the value of x, use the equation PS = PT:

`\overline{PS} = \overline{PT}`

`2x = 9`

`x = (9)/(2)=4.5`

To calculate the perimeter of triangle PQR, sum the tangent segments.

`\begin{aligned}\textsf{Perimeter}&=\overline{PS}+\overline{PT}+\overline{QT}+\overline{QU}+\overline{RS}+\overline{RU}\\&=9+9+4+4+13+13\\&=52\end{aligned}`

Therefore, the perimeter of triangle PQR is 52 units.