Word Lesson: Circles - Segments from Secants and Tangents

By M Ransom

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

know the basic properties of the segments formed by secants and tangents.
solve basic linear and quadratic equations.

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)

Solving this equation for x we get

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?

Remembering that when a secant and a tangent meet at a point outside a circle the product of the exterior part of the secant with its entire length is equal to the square of the tangent segment, we can generate the equation

PC² = PB(PB + BD)

This gives us

(x - 8)² = 4(4 + 12) = 64

Expanding, we have

x² - 16x + 64 = 64
x² - 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.

In circle O below (not drawn to scale), secants PA and PD are drawn from point P. We have the following measurements given: PC = 6, CA = 8, PB = x + 2, and BD = 5x +7. What is the measure of chord DB?

In the diagram below (which is not drawn to scale), circle O has secant PD and tangent PC. The following measurements are given: PC = x + 3, PB = 2x - 2, and BD = x + 2. What is the length of secant PD?

Circle O has radius 4.5. A secant and a tangent are drawn from an external point P. The secant passes through the center of the circle, point O. The tangent segment has length x + 2, and the external segment of the secant has length x. What is the length of the tangent segment?

14/5
Two possible answers are 6 and 3
10

Two secants to a circle of radius 8 meet in point A outside the circle. The external lengths of the secants are x and 10 respectively. The secant with an external length of 10 passes through the center of the circle. The internal length of the other secant is x - 6. What is the length of the secant that has an external length of x?

no possible answer because x is negative
20
26

When two secants, or a secant and a tangent, are drawn to a circle from the same external point, one of the following two relationships exists. The first is between the products of the lengths of the external portion of the secant and the lengths of the entire secant. The second is between the square of the length of the tangent segment and the external portion of the secant and the length of the entire secant. One of these two products is always equal when the secants and/or tangents are drawn from the same external point.

These relationships allow an equation to be formed. Based upon given measurements, it is often possible to find the lengths of other parts of the secants or the tangent segment itself. In the cases we have shown, it is sometimes necessary to solve a quadratic equation in order to find these lengths.