Question

In ∆ABC, LABC = 90° and AB = BC.
P is the midpoint of BC. PQ is perpendicular to AC at Q.
Prove that: ar(∆ABC) = 8ar(∆PQC)​

Answer 1

Answer:

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Answer 2

Given : In ∆ABC, LABC = 90° and AB = BC.

P is the midpoint of BC. PQ is perpendicular to AC at Q.

To Find  :  Prove that: ar(∆ABC) = 8ar(∆PQC)​

Solution:

ΔABC   and Δ PQC

∠ABC = ∠PQC = 90°

∠C = ∠C    common

=> ΔABC  ≈ Δ PQC

AB/PQ = BC/QC = AC/PC

PC = BC/2  = AB/2  ∵  AB = BC

=> AB/ PQ  = AB/QC  =  AC/(AB/2)

=>  AB/ PQ  = AB/QC  =  2AC/AB

AC² = AB² + BC²  = AB²  + AB²  = 2AB²

=> AC = √2AB

AC/PC =   2AC/AB = 2 √2AB / AB = 2√2

ΔABC  ≈ Δ PQC

=> Ar (ΔABC)  / Ar  (Δ PQC)  =  ( AC/ PC)²

=>  Ar (ΔABC)  / Ar  (Δ PQC)  = (2√2)²

=>  Ar (ΔABC)  / Ar  (Δ PQC)  = 8

=> Ar (ΔABC) = 8 Ar  (Δ PQC)

QED

Hence proved

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