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PN is a secant intersecting a circle at points M and N such that PN > PM. A tangent PT is drawn to the circle touching it at T. If PM = 32 cm and PT = 40 cm, what is the length (in cm) of the chord MN?
36
50
20
18
18
Use the tangent-secant theorem PT2=PM×PN to find PN, then subtract PM from PN to get chord MN.
PM = 32 cm, PT = 40 cm, PN is a secant and PT is a tangent.
PT2=PM×PN
Use the tangent-secant theorem PT2=PM×PN to find PN, then subtract PM from PN to get chord MN.
Students often forget that PN is the entire secant length, and they calculate MN directly from PN without subtracting PM.
State Tangent-Secant Theorem
When a tangent segment and a secant segment are drawn to a circle from an exterior point, the square of the length of the tangent segment is equal to the product of the lengths of the external secant segment and the entire secant segment.
PT2=PM×PN
Substitute the Given Values
Given PT=40 cm and PM=32 cm. Substitute these values into the formula to find the total length of the secant segment PN.
402=32×PN
Calculate the Total Secant Length PN
Simplify the equation: 1600=32×PN, which gives PN=321600=50 cm.
PN=50 cm
Find the Length of the Chord MN
Since P,M,N are collinear along the secant line in that order (with PM<PN), the length of chord MN is simply the difference between PN and PM.
MN=PN−PM=50−32=18 cm
D is correct because the calculated length of the chord MN is 18 cm.
The tangent-secant theorem is a special case of the intersecting chords theorem (power of a point theorem), which frequently appears in advanced geometry problems involving circles.
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