Question

QT and RS are medians of a triangle PQR right angle at p prove that A(QT^2+RS^2)=5QR^2

Answer 1

Triangle PQR and right angle at P

QT and RS are medians

Therefore, 2SP=PQ and 2TP=PR

It created two triangles which are tri PRS and tri QTP, then apply hypotenuse theorem that defined as square of hypotenuse side is equal to square of right angle side plus square of base side.  

By solving both triangles, you will get two equations. By adding these equations, A(QT^2+RS^2)=5QR^2 will satisfy.  

Answer 2

Answer:

4(SR² + QT²) = 5QR²

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

Hope, it helps !