SOLUTION: In ∆ PQR, │PQ│= 9cm, │QR│= 11cm, │RP│= 7cm and the bisector of ln ∠P meets line QR at T. Calculate │QT│ and│TR│.

Algebra ->  Test -> SOLUTION: In ∆ PQR, │PQ│= 9cm, │QR│= 11cm, │RP│= 7cm and the bisector of ln ∠P meets line QR at T. Calculate │QT│ and│TR│.      Log On


   

Question 1210339: In ∆ PQR, │PQ│= 9cm, │QR│= 11cm, │RP│= 7cm and the bisector of ln ∠P meets line QR at T. Calculate │QT│ and│TR│.
Answer by ikleyn(52747) About Me  (Show Source):
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In ∆ PQR, │PQ│= 9cm, │QR│= 11cm, │RP│= 7cm and the bisector of ln ∠P meets line QR at T.
Calculate │QT│ and│TR│.
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In every triangle, every its angle bisector has this remarkable property:


    the foot of the angle bisector divides the side of the triangle 
    in the ratio equal to the ratio of the adjacent sides.


Therefore, in triangle PQR, the bisector PT of angle P
divide the side QR of 11 cm long in the ratio

    abs%28QT%29%2Fabs%28TR%29 = abs%28PQ%29%2Fabs%28PR%29,    (1)

or

    abs%28QT%29%2Fabs%28TR%29 = 9%2F7.    (2)


Together with  QT + TR = QR = 11 cm,  it gives


    QT = 11%2A%289%2F%289%2B7%29%29 = 99%2F16 cm = 63%2F16 cm,    (3)

    TR = 11%2A%287%2F%289%2B7%29%29 = 77%2F16 cm = 413%2F16 cm.   (4)


ANSWER.  QT = 11%2A%289%2F%289%2B7%29%29 = 99%2F16 cm = 63%2F16 cm;  TR = 11%2A%287%2F%289%2B7%29%29 = 77%2F16 cm = 413%2F16 cm.

Solved.

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Try to deduce formulas (3) and (4) from formula (2) on your own.