Circles: Chords and Angles

Introduction: A circle is all points equidistant from one point called the center of the circle. Segments drawn within, through, or tangent to the circle create angles which we can define and measure.

Important facts:

The measure of a central angle is the same as the measure of the intercepted arc.
The measure of an inscribed angle is half the measure of the intercepted arc.
A segment connecting two points on a circle is called a chord.
A line passing through two points on a circle is called a secant.
A line external to a circle, passing through one point on the circle, is a tangent.

The Lesson:

We show circle O below in Figure A.

Points A, C, B, D, and P are on the circle.
The segments AB and CD are chords.
We have also drawn segments AC and BD to form triangles ACQ and DBQ.

In Figure B we show just the chords AB and CD and the triangles they form: ACQ and DBQ.

Figure A	Figure B

ANGLES formed by chords in a circle:

Figure A

Using Figure A, since angles 2 and 3 are inscribed angles

Secondly, we note that angle 1, formed at the intersection of the two chords AB and CD, is an exterior angle with respect to angles 2 and 3 in triangle ACQ.

The theorem from geometry states that an exterior angle of a triangle is equal in measure to the sum of the two remote interior angles. Therefore

SEGMENTS formed by intersecting chords in a circle:

Figure B

Using Figure B, we can examine the triangles ACQ and DBQ.

Because they are inscribed angles intercepting the same arcs, we have

This means that all three angles of the triangles ACQ and DBQ are equal in the correspondence:

A~D, C~B, and of course Q~Q

We now know that DACQ ~ DDBQ,
Since these triangles are similar we know that the following proportion is true:

ab = cd

In Summary:

The angle formed by intersecting chords is equal to ½ the sum of the intercepted arcs.
The products of the segments formed by intersecting chords are equal.

Let's Practice:

In the diagram shown below, circle O is given with and . What is the measure of angle 1?

Since angle 1 is formed by intersecting chords, we have:

The diagram given below shows circle O with chords AB and CD. CD is divided into segments of lengths x and 4x. AB is divided into segments of lengths 2x and 5. What is the measure of chord CD?

The products of the segments forming the chords are equal.

x(4x) = 2x(5)
4x² = 10x

4x² - 10x = 0
x(4x - 10) = 0
x = 0 or 4x = 10 x = 5/2

We will not use x = 0 since chord CD would also equal 0.

If we use x = 5/2, chord CD would measure

CD = x + 4x
CD = 5/2 + 4(5/2)
CD = 25/2

Examples

	In the circle O shown below, we have central angle COA with a measure of 120° and arc AB is 40°. What is the measure of angle DPB formed by Chords CB and AD?

	What is your answer?

	In the circle O shown below, chords CB and AD intersect at point P. We also know that CP = 7, BP = 2x + 1, AP = 3x – 1, and DP = 5. How long is chord CB?

	What is your answer?