Question

In given figure, is a right triangle PQR, right angled at Q. X and Y are the points on PQ and QR such that PX : XQ ï€ 1 : 2 and QY : YR ï€ 2 : 1. Prove that 9(PY2XR2)ï€13PR2

Answer 1

Given that X divides PQ in the ratio 1:2
QX/PQ = ⅔
QX = (⅔)PQ………..(1)
Y divides QR in the ratio 2:1
YQ/QR= ⅔
YQ = (⅔) RQ………..(2)
In right angled ∆PQY
PY² = YQ² + PQ² [By Pythagoras Theorem]
PY² = (⅔RQ)² + PQ² (From eq 2)
PY² = 4/9RQ² +PQ²
PY² = (4RQ² +9PQ²)/9
9PY² = 4RQ² +9PQ²………..(3)
In right angled ∆RQX
XR² = RQ² + QX² [By Pythagoras Theorem]
XR² = RQ² + (2/3PQ)² (From eq 2)
XR² = RQ² +4/9PQ²
XR² = (9RQ² +4PQ²)/9
9XR² = 9RQ² +4PQ²………..(4)
In ∆PQR
PR² = RQ² +PQ²…………….(5)
[By Pythagoras Theorem]
On Adding eq 3 &4
9PY² + 9XR²= 4RQ² +9PQ²+9RQ² +4PQ²
9(PY² + XR²) = 4RQ² +9RQ² +9PQ²+4PQ²
9(PY² + XR²) =13RQ² +13PQ²
9(PY² + XR²) =13(RQ² +PQ²)
9(PY² + XR²) =13PR² (From eq.5)

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