# Circles

### Topics

## Introduction to :

Fun activity time! You'll need a hula-hoop and several huge rubber bands big enough to stretch all the way across the hula-hoop. If you don't have those things, that's fine too. You can just imagine them. Usually you have to buy those industrial-sized rubber bands by the pallet.

Given those materials, how would you go about building a musical instrument? Banging the hula-hoop on the driveway and calling it a "percussion instrument" doesn't count.

You could build a very simple harp or lyre-like instrument by stringing a rubber band across the center of the circle, like this:

Pluck that rubber band, and you've got a note. Congratulations! You can now play any piece of music ever written…as long as it's Mozart's *Monotone Concerto*…which doesn't actually exist.

If you want to play more than one pitch, you'll need more rubber bands. You'll have to stretch them across the circle at different places so that the rubber bands end up being stretched to different lengths. Like this.

In geometry, we call objects like these rubber bands **chords**. No, we're not lyre-ing.

A line segment is a **chord of ⊙ O **if both the segment's endpoints are on ⊙

*O*. A chord that goes through the center of the circle is special, so we give it a special name: the diameter. A chord of ⊙

*O*is a

**diameter of ⊙**if it contains

*O**O*.

The length of a diameter of a circle is called "the diameter of the circle," just as the length of a radius of a circle is called "the radius of the circle." Hopefully that doesn't annoy you. If it does, blame the ancient Greeks. Yes, we know that's a bit of a cop-out. Blame the ancient Greeks for that too.

### Sample Problem

How many chords are in the following figure? How many diameters?

First, let's find all the chords. Any segment whose endpoints are on ⊙*O* are chords. Just looking at the figure, we can see that we have 5 chords: *AG*, *AE*, *BF*, *CG*, and *DF*. Diameters are chords that go through the center, which means we can have a maximum of 5 diameters. In this case, we only have 2: *AE* and *BF*. While *OC* goes through point *O*, it is not a diameter because only one of its endpoints is on ⊙*O* (the other is *O*, the center, which makes it a radius).

#### Example 1

Any diameter is also a chord. True or false? |

#### Example 2

Any radius is also a chord. True or false? |

#### Example 3

In the figure below, points |

#### Example 4

In the figure below, points |

#### Exercise 1

Refer to the figure below. Given that *AC* and *BD* are diameters of ⊙*O*, prove that *AB* is congruent to *CD*.

#### Exercise 2

Suppose *AB* is a chord of ⊙*O* and *CD* is a chord of ⊙*P*. Is it true that if *AB* = *CD*, then ⊙*O* is congruent to ⊙*P*?

#### Exercise 3

Given a circle with radius *r*, what is the maximum length of a chord of that circle?

#### Exercise 4

All chords are diameters, but not all diameters are chords. True or false?

#### Exercise 5

If two chords in the same circle are not congruent, then their associated arcs are also not congruent. True or false?