### 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. Recall that the product of the outer portion of a secant and the whole is equal to the same of the other secant. If one of these segments is a tangent, it will still be the product of the outer portion and the whole. However, for a tangent line, the outer portion and the whole are equal.

**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 to the left), then \begin{align*}a^2=b(b+c)\end{align*}. This means that the product of the outside segment of the secant and the whole is equal to the square of the tangent segment.

#### Finding the Value of a Missing Segment

1. Find the value of the missing segment.

Use the Tangent Secant Segment Theorem. Square the tangent and set it equal to the outer part times the whole secant.

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

2. Find the value of the missing segment.

Use the Tangent Secant Segment Theorem. Square the tangent and set it equal to the outer part times the whole secant.

\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*}

#### Solving for Unknown Values

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*}

### Examples

Find \begin{align*}x\end{align*} in each diagram below. Use the Tangent Secant Segment Theorem and simplify any radicals.

#### Example 1

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

#### \begin{align*}x^2&=16(16+25)\\x^2&=656\\x&=4\sqrt{41}\end{align*}Example 3

\begin{align*}x^2&=24(24+20)\\x^2&=1056\\x&=4\sqrt{66}\end{align*}

### Review

Solve for the missing segment.

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.

- Find \begin{align*}x\end{align*} and \begin{align*}y\end{align*}.

### Review (Answers)

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