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PQR is a triangle such that PQ = PR. RS and QT are the median to the sides PQ and PR respectively. If the medians RS and QR intersect at right angle, then prove that (PQ/QR) 2 = 5/2.

PQR is a triangle such that PQ = PR. RS and QT are the median to the sides PQ and PR respectively. If the medians RS and QR intersect at right angle, then prove that (PQ/QR)2 = 5/2.

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Grade:11

1 Answers

Arun
25750 Points
4 years ago
Step-by-step explanation:
Hi,
Consider Δ PQR , since ∠P = 90°,
Using Pythagoras Theorem,
QP² + RP² = QR² --------(1)
Consider Δ PSR , since ∠P = 90°,
Using Pythagoras Theorem,
SP² + RP² = SR² --------(2)
Consider Δ PQT , since ∠P = 90°,
Using Pythagoras Theorem,
QP² + TP² = QT² --------(3)
Adding (2) and (3), we get
SR² + QT² = SP² + RP² + QP² + TP²
Using (1), we get
SR² + QT² = QR² + SP² + TP²
But SP = 1/2*PQ (Since RS is the median)
TP = 1/2*PR ( Since QT is the median)
Thus,
SR² + QT² = QR² + 1/4* RP² +1/4*QP²
= QR² + 1/4*(RP² + QP²)
Again Using (1), we get
SR² + QT² = 5/4*QR²
or 
4(SR² + QT²) = 5QR²
 
 

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