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.
Please give proper answer with steps and no scam.
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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.
Please give proper answer with steps and no scam.
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Verified answer
Given :-
To find :-
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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