Question

show that the ratio of areas of two similar triangles is equal to the ratio of the squares of their corresponding sides.

Answer 1

Answer:

Given: Δ ABC ~ Δ PQR

To Prove: ar(ΔABC) / ar(ΔPQR) = (AB/PQ)2 = (BC/QR)2 = (CA/RP)2

Construction: Draw AM ⊥ BC, PN ⊥ QR

ar(ΔABC) / ar(ΔPQR) = (½ × BC × AM) / (½ × QR × PN)

= BC/QR × AM/PN ... [i]

In Δ ABM and Δ PQN,

∠B = ∠Q (Δ ABC ~ Δ PQR)

∠M = ∠N (each 90°)

So, Δ ABM ~ Δ PQN (AA similarity criterion)

Therefore, AM/PN = AB/PQ ... [ii]

But, AB/PQ = BC/QR = CA/RP (Δ ABC ~ Δ PQR) ... [iii]

Hence, from eqn.(i) :

ar(ΔABC) / ar(ΔPQR) = BC/QR × AM/PN

= AB/PQ × AB/PQ [From eqn.(ii) and (iii)] :

= (AB/PQ)2

Using eqn.(iii) :

ar(ΔABC) / ar(ΔPQR) = (AB/PQ)2 = (BC/QR)2 = (CA/RP)2

i hope it will helps you friend

Answer 2

Answer


Given: Δ ABC ~ Δ PQR

To Prove: ar(ΔABC) / ar(ΔPQR) = (AB/PQ)2 = (BC/QR)2 = (CA/RP)2

Construction: Draw AM ⊥ BC, PN ⊥ QR

ar(ΔABC) / ar(ΔPQR) = (½ × BC × AM) / (½ × QR × PN)

= BC/QR × AM/PN ... [i]

In Δ ABM and Δ PQN,

∠B = ∠Q (Δ ABC ~ Δ PQR)

∠M = ∠N (each 90°)

So, Δ ABM ~ Δ PQN (AA similarity criterion)

Therefore, AM/PN = AB/PQ ... [ii]

But, AB/PQ = BC/QR = CA/RP (Δ ABC ~ Δ PQR) ... [iii]

Hence, from eqn.(i)
:

ar(ΔABC) / ar(ΔPQR) = BC/QR × AM/PN

= AB/PQ × AB/PQ [From eqn.(ii) and (iii)]
:

= (AB/PQ)2

Using eqn.(iii)
:

ar(ΔABC) / ar(ΔPQR) = (AB/PQ)2 = (BC/QR)2 = (CA/RP)2