What if Ishmael wanted to know the diameter of a CD from his car? He found a broken piece of one in his car and took some measurements. He places a ruler across two points on the rim, and the length of the chord is 9.5 cm. The distance from the midpoint of this chord to the nearest point on the rim is 1.75 cm. Find the diameter of the CD.

### Segments from Chords

When two chords intersect inside a circle, the two triangles they create are similar, making the sides of each triangle in proportion with each other. If we remove and the ratios between , and will still be the same.

**Intersecting Chords Theorem:** If two chords intersect inside a circle so that one is divided into segments of length and and the other into segments of length and then . In other words, the product of the segments of one chord is equal to the product of segments of the second chord.

#### Solving for Unknown Values

1. Find in the diagram below.

Use the ratio from the Intersecting Chords Theorem. The product of the segments of one chord is equal to the product of the segments of the other.

2. Find in the diagram below.

Use the ratio from the Intersecting Chords Theorem. The product of the segments of one chord is equal to the product of the segments of the other.

3. Solve for .

a)

Again, we can use the Intersecting Chords Theorem. Set up and equation and solve for .

b)

However, because length cannot be negative, so .

#### CD Problem Revisited

Think of this as two chords intersecting each other. If we were to extend the 1.75 cm segment, it would be a diameter. So, if we find in the diagram below and add it to 1.75 cm, we would find the diameter.

### Examples

Find in each diagram below. Simplify any radicals.

For all problems, use the Intersecting Chords Theorem.

#### Example 1

#### Example 2

#### Example 3

### Review

Answer true or false.

- If two chords bisect one another then they are diameters.
- Tangent lines can create chords inside circles.
- If two chords intersect and you know the length of one chord, you will be able to find the length of the second chord.

Solve for the missing segment.

Find in each diagram below. Simplify any radicals.

Find the value of .

- Suzie found a piece of a broken plate. She places a ruler across two points on the rim, and the length of the chord is 6 inches. The distance from the midpoint of this chord to the nearest point on the rim is 1 inch. Find the diameter of the plate.
- Prove the Intersecting Chords Theorem.

Given: Intersecting chords and .

Prove:

### Review (Answers)

To view the Review answers, open this PDF file and look for section 9.9.