Q. 244.4( 14 Votes )

Prove that the ra

Answer :

We have


Given: ∆ABC ∆PQR


To prove: Ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides.


i.e.,


Construction: Draw AM perpendicular to BC and PN perpendicular to QR.


Proof: We know area of the triangle is given by (1/2 × base × height).


So in ∆ABC,


Area (∆ABC) = 1/2 × BC × AM [ base = BC & height = AM] …(i)


Similarly, in ∆PQR


Area (∆PQR) = 1/2 × QR × PN [ base = QR & height = PN] …(ii)


Dividing equations (i) by (ii), we get



…(iii)


In ∆ABM & ∆PQN,


B = Q [ ∆ABC ∆PQR; and corresponding angles are equal of similar triangles]


AMB = PNQ [ they are 90°]


So by AA-similarity property of triangle, ∆ABM ∆PQN.


[ corresponding sides of similar triangles are proportional] …(iv)


Substituting equation (iv) in equation (iii), we have



…(v)


We know that,


∆ABC ∆PQR


Using property which says that corresponding sides of similar triangles are proportional, we can write as



Using this equality, re-write equation (v)




…(A)


Using this equality again, re-writing equation (v)




…(B)


Similarly, …(C)


Collecting equations (A), (B) & (C), we get



Hence, proved.

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