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:

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?

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.

Let's Practice

Question #1

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?

Question #2

In the diagram below (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?

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?

A. no possible answer because x is negative

B. 20

C. 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.