Question

If two triangles are similar, then the ratio of their areas will equal to ratio of the corresponding sides.

(a) True

(b) False

I had been asked this question in an online interview.

This interesting question is from Area of Similar Triangle in section Triangles of Mathematics – Class 10

Answer 1

Right choice is (b) False

Easiest explanation: Ratio of similar triangles is equal to the square of their sides.

Since, ∆ABC ∼ ∆PQR, so they are equiangular and their sides are proportional.

∴ \(\frac {AB}{PQ}=\frac {AC}{PR}=\frac {BC}{QR}\), ∠A = ∠P, ∠B = ∠Q, ∠C = ∠R

\( \frac {area \, of \, triange \, ABC}{area \, of \, triange \, PQR}=\frac {\frac {1}{2} \times BC \times AD}{\frac {1}{2} \times QR \times PS}=\frac {BC \times AD}{QR \times PS}\)

Now, ∆ADB and ∆PS,

∠ABD = ∠PQS

∠B = ∠Q

Hence, ∆ABC ∼ ∆PQS

So, \( \frac {AD}{PS}=\frac {AB}{PQ}\)

But, \( \frac {AB}{PQ}=\frac {BC}{QR}\)

∴ \( \frac {AD}{PS}=\frac {BC}{QR}\)

\( \frac {area \, of \, triange \, ABC}{area \, of \, triange \, PQR}=\frac {\frac {1}{2} \times BC \times AD}{\frac {1}{2} \times QR \times PS}=\frac {BC^2}{QR^2}\)