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? After completing this Concept, you'll be able to use the Tangent Secant Segment Theorem to solve problems like this one.

### Watch This

Segments from Secants and Tangents CK-12

### Guidance

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*}.

#### Example A

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 B

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 C

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*}

Segments from Secants and Tangents CK-12

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### Guided Practice

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

1.

2.

3.

**Answers:**

Use the Tangent Secant Segment Theorem.

1.

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

2.

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

3.

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

### Explore More

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.