InterviewSolution
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InterviewSolution
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Let PQR be am acute-angled triangle in which PQ lt QR. From the vertex Q draw altitude `QQ_(1)`, the angle bisector `Q Q_(2)` and the median`Q Q_(3)" With "Q_(1),Q_(2),Q_(3)` lying on PR. Then A. `PQ_(1)ltPQ_(2)ltPQ_(3)`B. `PQ_(2)ltPQ_(1)ltPQ_(3)`C. `PQ_(1)ltPQ_(3)ltPQ_(2)`D. `PQ_(3)ltPQ_(1)ltPQ_(2)` |
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Answer» Correct Answer - A `PQ_(3)=Q_(3)R( :. Q Q_(3)" ie median")` `PQ_(3)=1/2PR` `PQ_(2):Q_(2)R=r:p" (By peoperty of angle bisector)"` `PQ_(2)=(r/(r+P))PR` But r lt P (Given) `PQ_(2)lt1/2PR` Comparison between Altitude and angle bisector `rArr angleQPQ_(2)+anglePQ_(2)Q+anglePQ Q_(2)=angle RQ Q_(2)+angleQ Q_(2)R+angleQRQ_(2)` `:. anglePQ Q_(2)+angleRQ Q_(2)" {Since angle bisector}"` `angle QPQ_(2)+anglePQ_(2)Q=angle Q Q_(2)R+angleQRQ_(2)` `:.PQltQR" then"ltQPQ_(2)gtangle QRQ_(2)` Hence `angleQ Q_(2)P+ltangleQ Q_(2)R` But `angle Q Q_(2)P+angleQ Q_(2)R=180^(@)` Hence `angle Q Q_(32)Plt90^(@)&angle Q Q_(2)Rgt 90^(@)` `rArr" Foot from Q to side PR lies inside "DeltaPQ Q_(2)` `rArrPQ_(1)lt PQ_(2)ltPQ_(3)` |
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