Question

if triangle ABC is similar to triangle PQR and AD and PS are the bisectors of corresponding angles A and P then prove that area of triangle ABC / area of triangle PQR = AD2/PS2

Answer 1

Given : ΔABC is similar to triangle PQR and AD and PS are the bisectors of corresponding angles A and P then

To Find :  prove that area of triangle ABC / area of triangle PQR = AD²/PS²

Solution:

ΔABC ≈ Δ PQR

ratio of the area of two similar triangles =( ratio of their corresponding sides)²

Ar ΔABC / Ar Δ PQR   = (AB / PQ)²

ΔABC ≈ Δ PQR

=> ∠A = ∠P  

  ∠B = ∠Q

    ∠A = ∠P  

=> ∠A/2 = ∠P/2  

=> ∠BAD = ∠QPS

in  ΔABD and  Δ PQS

 ∠BAD = ∠QPS

  ∠B = ∠Q

=>  ΔABD ≈ Δ PQS

=> AB/PQ = AD/PS

Ar ΔABC / Ar Δ PQR   = (AB / PQ)²

=> Ar ΔABC / Ar Δ PQR   = (AD/PS)²

QED

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