Side Length of Tangent & Secant of a Circle

Tangents, secants, Side Lengths Theorems & Formula

If a tangent and a secant or if two secants intersect from a point outside of the circle, then there are two useful theorems/formula that relate the side lengths of the two given segments (either tangent & secant or secant & secant)

Tangent and Secants

If a secant and a tangent of a circle are drawn from a point outside the circle, then the product of the lengths of the secant and its external segment equals the square of the length of the tangent segment.

Picture of tangent secant and sides

Practice Problems

Lengths of secant and a tangent

Problem 1

Use the theorem for the intersection of a tangent and a secant of a circle to solve the problems below.

In the diagram on the left, the red line is a tangent, how long is it?

picture of intersection of  secant and a tangent

x² = (7+5) • 5
x² = (12) • 5
x² = 60
x =

Problem 2

In the problem below, the red line is a tangent of the circle, what is its length?

x² = (7+9) • 7
x² = (16) • 7
x² = 112
x =

Two Secants Intersecting

If two secant segments are drawn from a point outisde a circle, the product of the lengths(C+D) of one secant segment and its exteranal segment(D) equals the product of the lengths (A+B) of the other secant segment and its external segment (B).

Product of segments of secants from point out of circle
Problem 3

Use the theorem above to determine A if B = 4, C = 8, D = 5

(A +4) • 4 = (5 +8) • 5
(A +4) • 4 = (13) • 5
(A +4) • 4 = 65
(A +4) = 65 ÷4
(A +4) = 16.25
A = 16.25 − 4
A = 12.25

Problem 4

Use the theorem above to determine A if B = 8, C = 16, D = 10

practice finding secant length

(A +8) • 8 = (16 +10) •10
(A +8) • 8 = (26) • 10
(A +8) • 8 = 260
(A +8) = 2,60÷ 8
(A +8) = 32.5
A = 32.5 − 8
A = 24.5

Problem 5

The two secants in the picture below are not drawn to scale. If KO = 16, KJ =4, and LO = 32, what is the measure of LM?

practice problem two secants side length formula

How to use the theorem to find side LM

KOJO = LO MO
JO = KOKJ
JO = 16 − 4 = 12
KOJO = LO MO
16 • 12 = 32 • MO
192= 32 • MO
192/32= MO
6 = MO
LM = LOMO
LM = 32 − 6 =26