Question

If each side of a scalene triangle is tripled then prove that ratio of their area will be 1:9.

Answer 1

Let $ABC$ and $PQR$ be the two triangles.

To prove:- $\frac{ar\left(ABC\right)}{ar\left(PQR\right)}=\frac{1}{9}$

Proof:-

Let the side of first triangle $ABC$ be $a,b$ and $c$.

Therefore,

Sides of second triangle $PQR$ will be $3a,3b$ and $3c$

Now,

$\frac{AB}{PQ}=\frac{BC}{QR}=\frac{CA}{PR}=\frac{1}{3}$

Therefore, by S.S.S. similarity,

$\therefore △ABC\sim △PQR$

Since ratio of area of similar triangles is equal to the ratio of squares of their corresponding sides.

Therefore,

$\frac{ar\left(ABC\right)}{ar\left(PQR\right)}=\frac{{AB}^2}{{PQ}^2}=\frac{a^2}{{\left(3a\right)}^2}$

$\frac{ar\left(ABC\right)}{ar\left(PQR\right)}=\frac{1}{9}$

Hence proved.