Question

∆ABC मध्ये रेषा बाजू ABव AC ला
अनुक्रमे बिंदू P व Qछेदते . तर सिद्ध करा
A (∆APQ)=AP×AQ
A(∆ABC)=AB×AC
(सूचना : बिंदू B व Q जोडा)​

Answer 1

Given :   PQ join AB and AC   at mid point

रेषा PQहीABC च्या बाजू AB बाजू AC ला अनुक्रमे P व Q मध्ये छेदते

To Find : Prove that

Ar Δ APQ / Ar ΔABC  =  AP * A Q / AB * AC

Solution:

line joining the mid-point of two sides of a triangle is equal to half the length of the third side and parallel to third side

PQ || BC

ΔAPQ and ΔABC

∠A = ∠A  common

∠P  = ∠B      (corresponding angles )

∠Q  = ∠C      (corresponding angles )

=> ΔAPQ ≈ ΔABC

AP/AB = AQ/AC  = PQ/BC

Ratio of area of similar triangles = square of ratio of corresponding sides

Ar  ΔAPQ /  Ar ΔABC  = ( AP/ AB)²

=> Ar  ΔAPQ /  Ar ΔABC  = ( AP/ AB)  ( AP/ AB)

AP/AB = AQ/AC

=> Ar  ΔAPQ /  Ar ΔABC  = ( AP/ AB)  ( AQ/ AC)

=> Ar Δ APQ / Ar ΔABC  =  AP * A Q / AB * AC

QED

Hence proved

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