### Segments from Secants

When two secants intersect outside a circle, the circle divides the secants into segments that are proportional with each other.

**Two Secants Segments Theorem:** If two secants are drawn from a common point outside a circle and the segments are labeled as below, then \begin{align*}a(a+b)=c(c+d)\end{align*}.

### Examples

#### Example 1

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

Use the Two Secants Segments Theorem.

\begin{align*}8(8+x)&=6(6+18)\\64+8x&=144\\8x&=80\\x&=10\end{align*}

#### Example 2

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

Use the Two Secants Segments Theorem.

\begin{align*}15(15+27)&=x\cdot 45\\630&=45x\\x&=14\end{align*}

#### Example 3

Find the value of \begin{align*}x\end{align*}.

Use the Two Secants Segments Theorem.

\begin{align*}18 \cdot (18+x) &= 16 \cdot (16+24)\\ 324+18x &= 256+384\\ 18x &= 316\\ x &= 17\frac{5}{9}\end{align*}

#### Example 4

Find the value of \begin{align*}x\end{align*}.

Use the Two Secants Segments Theorem.

\begin{align*}x \cdot (x+x) &= 9 \cdot 32\\ 2x^2 &= 288\\ x^2 &= 144\\ x &=12, \ x\ne -12 \ (\text{length is not negative})\end{align*}

#### Example 5

True or False: Two secants will always intersect outside of a circle.

False. If the two secants are parallel, they will never intersect. It's also possible for two secants to intersect inside a circle.

### Review

Fill in the blanks for each problem below. Then, solve for the missing segment.

\begin{align*}3(\underline{\;\;\;\;\;\;\;}+\underline{\;\;\;\;\;\;\;})=2(2+7)\end{align*}

\begin{align*}x \cdot \underline{\;\;\;\;\;\;\;}=8(\underline{\;\;\;\;\;\;\;}+\underline{\;\;\;\;\;\;\;})\end{align*}

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

- Fill in the blanks of the proof of the Two Secants Segments Theorem.

Given: Secants \begin{align*}\overline{PR}\end{align*} and \begin{align*}\overline{RT}\end{align*}

Prove: \begin{align*}a(a+b)=c(c+d)\end{align*}

Statement |
Reason |
---|---|

1. Secants \begin{align*}\overline{PR}\end{align*} and \begin{align*}\overline{RT}\end{align*} with segments \begin{align*}a, \ b, \ c,\end{align*} and \begin{align*}d\end{align*}. | 1. Given |

2. \begin{align*}\angle R \cong \angle R\end{align*} | 2. Reflexive PoC |

3. \begin{align*}\angle QPS \cong \angle STQ\end{align*} | 3. Congruent Inscribed Angles Theorem |

4. \begin{align*}\triangle RPS \sim \triangle RTQ\end{align*} | 4. AA Similarity Postulate |

5. \begin{align*}\frac{a}{c+d}=\frac{c}{a+b}\end{align*} | 5. Corresponding parts of similar triangles are proportional |

6. \begin{align*}a(a+b)=c(c+d)\end{align*} | 6. Cross multiplication |

Solve for the unknown variable.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 9.10.