In order to solve problems which involve intersecting chords in circles, it is necessary to

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

A typical problem involving the angles and segments formed by intersecting chords in a

circle gives us information about the lengths of parts of the chords, about the angles formed by the chords, and/or about the arcs of the

circle intercepted by these angles. Two examples of this type of problem follow:

- In circle O shown below, chords CB and AD intersect at point P. The segments formed by these intersecting chords are CP = 7, BP = x, AP = 2x, and DP = x + 1. What is the measure of chord CB?

We note that because the chords intersect, we have

(CP)(BP) = (AP)(DP)

7x = 2x(x + 1)

7x = 2x^{2} + 2x

To solve for x, we will collect like terms and set our

equation equal to zero.

2x^{2} - 5x = 0

x(2x - 5) = 0

x = 0 or x =5/2

Although x = 0 is an answer, this would make BP = 0. We will use x = 5/2.

CB = CP + BP

CB = 7 + x

CB = 7 + 5/2

CB = 19/2

- In circle O given below, suppose that angle 3 is 40° and angle B is 100°. What is the measure of angle 1?

Notice that the

arc CB is intercepted by

angle 3.

Since

angle 3 is 40°, we know that

arc CB is 80°.

Similarly, we know that

arc APD is 200° since

angle B is 100°

Since

angle 1 is formed by intersecting chords, it has a measure equal to one half the sum of the intercepted arcs CB and APD.