Theorem 6.6 Class X

Theorem 6.6

Statement

"The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides"

Given:-

Two similar triangles ABC and PQR are given.

To prove that:-

ar(ABC)/ar(PQR) = (AB/PQ)² = (BC/QR)² = (AC/PR)²

Construction:-

Draw AM and PN perpendicular on BC and QR respectively.

Proof:-

ar(ABC) = BCxAM/2

ar(PQR) = QRxPN/2

Therefore,

ar(ABC)/ar(PQR) = BCxAM/QRxPN .............................. (i)
Now, consider in triangles ABM and PQN are similar under AAA similarity criterion.
<B = <Q { Triangles ABC and PQR are similar}
<M = <N {Each right angle}
<BAM = <QPN {Remain angle}

Therefore,
AB/PQ = AM/PN ............................. (ii)
Since, Triangles ABC and PQR are similar.

Now, from (i) and (ii);
ar(ABC)/ar(PQR) = BCxAB/QRxPQ ................................... (iii)

AB/PQ = BC/QR = AC/PR {Triangles ABC and PQR are similar}................ (iv)

Now, from (iii) and (iv)

ar(ABC)/ar(PQR) = (AB/PQ)² = (BC/QR)² = (AC/PR)² Proved