Question

triangle ABC~ Triangle PQR prove that ar(ABC)/ar(PQR)= (AB/PQ)²= (BC/QR)²= (CA/RP)²​

Answer 1

Answer:

$\mathfrak \pink \star \red{ answer}$

Proof: We are given two triangles ABC and PQR such that Δ ABC ~Δ PQR

We need to prove that ar(ABC) / ar(PQR)

= (AB/PQ)² = (BC/QR)²=(CA/RP)²

For Finding the areas of the two triangles, we draw altiude AM and PN of the triangle

Now, ar(ABC) = 1/2 BC x AM and ar(PQR) = 1/2 QR x PN

Therefore , AM/PN = AB/PQ

Also, Δ ABC ~ Δ PQR

So, AB/PQ = BC/QR = CA/RP

Therefore, ar(ABC)/ar(PQR) = AB/PQ x AM/PN = AB/PQ x AB/PQ = (AB/PQ)²

Weget ar(ABC)/ar(PQR) = (AB/PQ)² = (BC/QR)² = (CA/RP)²

So, ar(ABC)/ar(PQR) = (1/2 x BC x AM) / 1/2 x QR x PN = (BC x AM) / (QR x PN)

Now, in Δ ABM and Δ PQN, ∠B = ∠Q

(As Δ ABC ~ Δ PQR)

∠M = ∠N (Each is of 90⁰)

and Δ ABM ~Δ PQN (AA similarity criterion)