### Tangent Line Theorems

There are two important theorems about tangent lines.

1. **Tangent to a Circle Theorem:** A line is tangent to a circle if and only if the line is perpendicular to the radius drawn to the point of tangency.

is tangent at point if and only if .

This theorem uses the words “if and only if,” making it a biconditional statement, which means the converse of this theorem is also true.

2. **Two Tangents Theorem:** If two tangent segments are drawn to one circle from the same external point, then they are congruent.

and have as an endpoint and are tangent; .

What if a line were drawn outside a circle that appeared to touch the circle at only one point? How could you determine if that line were actually a tangent?

### Examples

#### Example 1

Determine if the triangle below is a right triangle.

Use the Pythagorean Theorem. is the longest side, so it will be .

Does

is not a right triangle. From this, we also find that is not tangent to .

#### Example 2

If and are the centers and is tangent to both circles, find .

and and by AA Similarity.

To find , use the Pythagorean Theorem.

To find , use similar triangles.

#### Example 3

is tangent to at point . Find . Reduce any radicals.

is tangent, so and a right triangle. Use the Pythagorean Theorem to find .

#### Example 4

Using the answer from Example A above, find in . Round your answer to the nearest hundredth.

#### Example 5

Find the perimeter of .

and . Therefore, the perimeter of .

is ** inscribed** in . A circle is inscribed in a polygon if every side of the polygon is tangent to the circle.

### Review

Determine whether the given segment is tangent to .

Find the value of the indicated length(s) in . and are points of tangency. Simplify all radicals.

and are points of tangency for and .

- Is ? Why?
- Find .
- Find .
- Find .
- Find and .

is inscribed in .

- Find the perimeter of .
- What type of quadrilateral is ? How do you know?
- Draw a circle inscribed in a square. If the radius of the circle is 5, what is the perimeter of the square?
- Can a circle be inscribed in a rectangle? If so, draw it. If not, explain.
- Draw a triangle with two sides tangent to a circle, but the third side is not.
- Can a circle be inscribed in an obtuse triangle? If so, draw it. If not, explain.
- Fill in the blanks in the proof of the Two Tangents Theorem.

Given: and with points of tangency at and . and are radii.

Prove:

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

1. | 1. |

2. | 2. |

3. and | 3. |

4. | 4. Definition of perpendicular lines |

5. | 5. Connecting two existing points |

6. and are right triangles | 6. |

7. | 7. |

8. | 8. |

9. | 9. |

- Fill in the blanks, using the proof from #21.
- is a _____________ (type of quadrilateral).
- The line that connects the ___________ and the external point __________ .

- Points and are points of tangency for the three tangent circles.
*Explain*why .

### Review (Answers)

To see the Review answers, open this PDF file and look for section 9.2.