Question

QT and RS are medians of a triangle PQR right angled at P. Prove that

A(QT2
+RS2) = 5QR2

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Answer 1

Answer:

proved

Step-by-step explanation:

Given QT and RS are medians of a triangle PQR right angled at P. Prove that

4(QT2  + RS2) = 5QR2

In triangle PRS, QT and RS are medians, P is right angled.

So RS^2 = PR^2 + PS^2

multiplying by 4 we get

4RS^2 = 4PR^2 + 4PS^2

           = 4PR^2 +  QP^2 (since 2 PS = QP)

In triangle QTP,

 QT^2 = PT^2  +  PQ^2

multiplying by 4 we get

4QT^2 = 4PT^2 + 4PQ^2

            = RP^2 + 4 PQ^2 (since RP = 2 PT)

4 RS^2 + 4 QT^2 = 4 RP^2 + PQ^2 + RP^2 + 4 QP^2

       5 RP^2 + 5 PQ^2 = 5(RP^2 + PQ^2)

5 QR^2(RP^2 + QP^2 = QR^2)

SO 4SR^2 + 4QT^2 = 5QR^2