### 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 .

### Examples

#### Example 1

Find . Simplify any radicals.

Use the Two Secants Segments Theorem.

#### Example 2

Find . Simplify any radicals.

Use the Two Secants Segments Theorem.

#### Example 3

Find the value of .

Use the Two Secants Segments Theorem.

#### Example 4

Find the value of .

Use the Two Secants Segments Theorem.

#### 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.

Find in each diagram below. Simplify any radicals.

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

Given: Secants and

Prove:

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

1. Secants and with segments and . | 1. Given |

2. | 2. Reflexive PoC |

3. | 3. Congruent Inscribed Angles Theorem |

4. | 4. AA Similarity Postulate |

5. | 5. Corresponding parts of similar triangles are proportional |

6. | 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.