In triangle pqr, QS bisects PR and T is the mid point of QS. If PT produced meets QR at U, then find the value of QU in terms of QR.
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Answers
Given :-
- QR bisects PR.
- T is the mid - point of QS.
To find :-
- the value of QU in terms of QR. ?
Solution :-
Since QR bisects PR.
→ PS = SR => PS : SR = 1 : 1.
Also, T is the mid - point of QS.
→ QT = TS => QT : TS = 1 : 1 .
Now, using MPT (Mass point geometry) we have ratios of two sides . we will distribute mass according to their ratio.
From image now :-
PS : SR = 1 : 1
=> Mass at P = Let 2unit.
Than,
=> Mass at R = Also 2 unit.
Now, Mass at S will be = sum of (P + R) = 2 + 2 = 4 unit.
Now, on line segment QS,
=> QT : TS = 1 : 1
And,
=> Mass at S = 4 .
Therefore,
=> Mass at Q = 4 ( 4 : 4 = 1 : 1)
Now, on line segment QR we have ,
=> Mass at Q = 4
=> Mass at R = 2
Therefore,
=> QU : UR = 2 : 4 = 1 : 2 .
Hence,
=> QU : QR = QU : (QU + UR)
=> QU : QR = 1 : (1 + 2)
=> QU : QR = 1 : 3
=> QU = (1/3)QR. (Ans.)
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