Title: READ: Segments of Chords, Secants, and Tangents
Course: Geometry (B)

Segments of Chords, Secants, and Tangents

Learning Objectives

- Find the lengths of segments associated with circles.

In this section we will discuss segments associated with circles and the angles formed by these segments. The figures below give the names of segments associated with circles.

Segments of Chords

Theorem If two chords intersect inside the circle so that one is divided into segments of length a and b and the other into segments of length b and c then the segments of the chords satisfy the following relationship:

This means that the product of the segments of one chord equals the product of segments of the second chord.

Proof

We connect points and and points and to make and .

Therefore, by the AA similarity postulate.

In similar triangles the ratios of corresponding sides are equal.

Example 1

Find the value of the variable.

Segments of Secants

Theorem If two secants are drawn from a common point outside a circle and the segments are labeled as below, then the segments of the secants satisfy the following relationship:

This means that the product of the outside segment of one secant and its whole length equals the product of the outside segment of the other secant and its whole length.

Proof

We connect points and and points and to make and .

Therefore, by the AA similarity postulate.

In similar triangles the ratios of corresponding sides are equal.

Example 2

Find the value of the variable.

Segments of Secants and Tangents

Theorem If a tangent and a secant are drawn from a point outside the circle then the segments of the secant and the tangent satisfy the following relationship

This means that the product of the outside segment of the secant and its whole length equals the square of the tangent segment.

Proof

We connect points and and points and to make and .

Therefore, by the AA similarity postulate.

In similar triangles the ratios of corresponding sides are equal.

Example 3

Find the value of the variable assuming that it represents the length of a tangent segment.

Lesson Summary

In this section, we learned how to find the lengths of different segments associated with circles: chords, secants, and tangents. We looked at cases in which the segments intersect inside the circle, outside the circle, or where one is tangent to the circle. There are different equations to find the segment lengths, relating to different situations.

The questions are for your own review. The answers are listed below to help you check your work and understanding.

Review Questions

1. Find the value of missing variables in the following figures:
2. A circle goes through the points and consecutively. The chords and intersect at . Given that and find ?
3. 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 found to be The distance from the midpoint of this chord to the nearest point on the rim is found to be Find the diameter of the plate.
4. Chords and intersect at . Given and find .

Review Answers

  14. or