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)

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Two Secants Two Tangents
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.
Practice Problems
Lengths of secant and a tangent
x² = (7+5) • 5
x² = (12) • 5
x² = 60
x =
x² = (7+9) • 7
x² = (16) • 7
x² = 112
x =

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Two Secants Two Tangents
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).
(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
(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
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

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Two Secants Two Tangents