### Segments from Secants and Tangents

If a tangent and secant meet at a common point outside a circle, the segments created have a similar relationship to that of two secant rays.

**Tangent Secant Segment Theorem:** If a tangent and a secant are drawn from a common point outside the circle (and the segments are labeled like the picture below), then \begin{align*}a^2=b(b+c)\end{align*}.

What if you were given a circle with a tangent and a secant that intersect outside the circle? How could you use the length of some of the segments formed by their intersection to determine the lengths of the unknown segments?

### Examples

#### Example 1

Find \begin{align*}x\end{align*}. Simplify any radicals.

Use the Tangent Secant Segment Theorem.

\begin{align*} 18^2&=10(10+x)\\324&=100+10x\\224&=10x\\x&=22.4\end{align*}

#### Example 2

Find \begin{align*}x\end{align*}. Simplify any radicals.

Use the Tangent Secant Segment Theorem.

\begin{align*}x^2&=16(16+25)\\x^2&=656\\x&=4\sqrt{41}\end{align*}

#### Example 3

Find the length of the missing segment.

Use the Tangent Secant Segment Theorem.

\begin{align*}x^2 &= 4(4+12)\\ x^2 &= 4 \cdot 16 = 64\\ x &= 8\end{align*}

#### Example 4

Fill in the blank and then solve for the missing segment.

\begin{align*}\underline{\;\;\;\;\;\;\;}=\underline{\;\;\;\;\;\;\;}(4+5)\end{align*}

\begin{align*}x^2&=4(4+5)\\x^2&=36\\x&=6\end{align*}

#### Example 5

Find the value of the missing segment.

Use the Tangent Secant Segment Theorem.

\begin{align*}20^2 &= y(y+30)\\ 400 &= y^2+30y\\ 0 &= y^2+30y-400\!\\ 0 &= (y+40)(y-10)\!\\ y &= \xcancel{-40},10\end{align*}

### Review

Fill in the blanks for each problem below and then solve for the missing segment.

\begin{align*}10^2=x(\underline{\;\;\;\;\;\;\;}+\underline{\;\;\;\;\;\;\;})\end{align*}

Find \begin{align*}x\end{align*} in each diagram below. Simplify any radicals.

- Describe and correct the error in finding \begin{align*}y\end{align*}. \begin{align*}10 \cdot 10&=y\cdot 15y\\ 100&=15y^2\\ \frac{20}{3}&=y^2\\ \frac{2\sqrt{15}}{3}&=y \quad {\color{red}\longleftarrow \ y} \ {\color{red}\text{is \underline{not} correct}}\end{align*}

Solve for the unknown variable.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 9.11.