Side Length of Tangent & Secant of a Circle

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.

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?

Length of Tangent

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

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

Length of tangent

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

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).

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

Answer

(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

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

Answer

(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

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?

Answer

How to use the theorem to find side LM
KO • JO = LO • MO
JO = KO − KJ
JO = 16 − 4 = 12
KO • JO = LO • MO
16 • 12 = 32 • MO
192= 32 • MO
192/32= MO
6 = MO
LM = LO − MO
LM = 32 − 6 =26