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 = 2x2 + 2x
To solve for x, we will collect like terms and set our equation
equal to zero.
2x2 - 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 is 40°, we know that arc
CB is 80°.
Similarly, we know that arc
APD is 200° since angle
B is 100°
1 is formed by intersecting chords, it has a measure equal to one half the sum of the intercepted arcs CB and APD.