Prove that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding medians

Question

Philoid · Accepted Answer

Let us assume two similar triangles as ΔABC ~ ΔPQR.

Let AD and PS be the medians of these triangles

Then, because ΔABC ~ΔPQR

.....(i)

∠A = ∠P, ∠B = ∠Q, ∠C = ∠R .......(ii)

Since AD and PS are medians,

BD = DC =BC/2

And, QS = SR =QR/2

Equation (i) becomes,

.......(iii)

In ΔABD and ΔPQS,

∠B = ∠Q [From (ii)]

And,

[From (iii)]

ΔABD ~ ΔPQS (SAS similarity)

Therefore, it can be said that

.......(iv)

From (i) and (iv), we get

And hence,