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Geometry: Internal Bisector of an Angle of a Triangle
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Internal Bisector of an Angle of a Triangle

The internal bisector of an angle of a triangle divides the side opposite to it, internally in the ratio of the sides containing the angle.

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Example: In fig. below, PM is the bisector of QPR. If PQ = 12 cm, PR = 16 cm and QR = 8 cm, find QM and MR.
Solution: Let QM = x cm. Then, MR = (8 - x)cm.
As PM is the bisector of P. So
    PQ/PR = QM/MR
12/16 = x/(8 - x)
3/4 = x/(8 - x)
24 - 3x = 4x
7x = 24
x = 24/7
x = 3.43 cm
QM = 3.43 cm
MR = (8 - x) = (8 - 3.43) = 4.57 cm. 
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