In order to solve problems which involve secants, tangents, and segments formed by them, it is necessary to

A typical problem involving the segments formed by secants and tangents in a

circle gives us information about the measures of the secants and tangent and/or the segments formed when they intersect each other and the circle. Two examples of this type of problem are presented below.

- In circle O below (not drawn to scale), two secants from point P intersect circle O such that arcs CP = 10, BP = 9, CA = 2x, and BD = 2x +3. What is the measure of segment AP?

The products of the external

segment and the entire

secant must be equal for both secants. We have:

CP(CP + CA) = BP(BP + BD)

10(2x + 10) = 9(2x + 12)

20x + 100 = 18x + 108

2x = 8

x = 4

Since AP equals 2x + 10

AP = 2(4) + 10

AP = 18

- In circle O below (not drawn to scale), a tangent and secant are drawn from point P. We are given the following measurements: PC = x - 8, PB = 4, and BD = 12. What is the length of segment PC?

PC^{2} = PB(PB + BD)

This gives us

(x - 8)^{2} = 4(4 + 12) = 64

Expanding, we have

x^{2} - 16x + 64 = 64

x^{2} - 16x = 0

Solving for x

x(x - 16) = 0

x = 0 and x = 16

When we check x = 16 we get

PC = x - 8

PC = 16 - 8

PC = 8

Note that we cannot use x = 0 since it would give us PC = -8 and the

length of a segment cannot be a negative number.