### Topics

## Introduction to :

Lines and circles tend to avoid each other, because they kind of freak each other out. Lines don't care for the weird curviness a circle has, while circles are mystified by the fact that a line goes on forever and forever and forever...

Anyway, circles and lines don't get along too well, but since they are at the mercy of whoever draws them (to some extent), they do have to interact sometimes.

Throw a line down onto a plane with a circle in it and one of three things will happen. If they are lucky, the line and the circle won't intersect at all. They'll each go about their business, perhaps warily eying each other, but not sharing any points.

You might end up with the line intersecting the circle at two points.

Or, once in a blue moon, you might see the line just "touching" the circle, intersecting it at exactly one point.

We have special names for these last two cases. If a line intersects a circle at two points, then the line is a **secant of **the circle. If a line intersects a circle at exactly one point, then the line is **tangent to** the circle. (You can think of a tangent line as just barely touching the circle.)

#### Example 1

Suppose line *m* intersects ⊙*X* at point *Z* and *m *is perpendicular to *XZ*. What is the maximum number of other points on ⊙*X* that *m* can intersect? | |

The Perpendicular Tangent Theorem tells us that in the situation described above, line *m* must be tangent to ⊙*X* at *Z*. By definition of tangent line, line *m *must intersect ⊙*X* in exactly one point. That means that the maximum number of points other than *P* on ⊙*X* that *m* can intersect is zero. | |

#### Example 2

Is there a maximum number of secant lines two circles can have in common? If so, what is it? (In other words, given two circles, how many lines *m *can you draw that *m* is a secant of both circles?) | |

In this case, it's best to draw a picture. Start with any two circles you like. Spend a few seconds drawing common secants and you will find that there is no maximum number of secant lines two circles can have in common. You could keep on drawing them for the rest of your life if you wanted to. | |

#### Example 3

In the figure below, segments *CA* and *CB* are tangent to ⊙*O* at points *A* and *B*, respectively. If *OA* = 8 cm and *AC* = 18 cm, what is the length of *BC*? | |

In the given situation, the Endpoint-Tangent Theorem applies. It tells us that *AC* is congruent to *BC*. Therefore *BC* = *AC* = 18 cm. Note that we didn't even need to know *OA*. | |

#### Exercise 1

How is it possible for two circles to have only two common tangents?

Hint

Circles can "protect" each other from tangent lines... the "closer" two circles are, the safer they are from tangent lines.

Answer

The circles must intersect at two points.

#### Exercise 2

How is it possible for two circles to have only one common tangent?

Hint

Circles can "protect" each other from tangent lines... the "closer" two circles are, the safer they are from tangent lines.

Answer

One circle must be inside the other and they must be tangent at some point.

#### Exercise 3

A diameter is a secant. True or false?

Hint

Read the definitions of each of those terms carefully.

#### Exercise 4

If a line *m* is perpendicular to the radius of a circle, then *m *is not a secant of the circle. True or false?

Hint

Tangents *must* be perpendicular to the radius of a circle. What about secants?

#### Exercise 5

In the figure below, segments *PB* and *PA* are tangent to ⊙*O* at *B* and *A*, respectively. Also, ∠*AOB* is a right angle. Prove that ∠*APB* is a right angle.

Hint

*PBOA* is a quadrilateral and therefore the measures of its angles must add up to 360°.

Answer

If we recognize that a tangent segment is part of a tangent line, by the Perpendicular Tangent Theorem, we know that *PB* is perpendicular to *OB*. Likewise, *PA* is perpendicular to *OA*. Therefore, ∠*PAO* and ∠*PBO* are both right angles. *PBOA* is a quadrilateral, so the measures of all its angles must add up to 360°. Since each angle in *PBOA* other than ∠*APB* is a right angle, ∠*APB* must also be a right angle.