Question

Sides of two similar triangles are in the ratio $4:9$. Areas of these triangles are in the ratio?

• A. $2:3$
• B. $4:9$
• C. $81:16$
• D. $16:81$

Answer 1

The correct option is D

$16:81$

The ratio of Areas of similar triangles is in the ratio

Explanation for the correct option:

The sides of two similar triangles are in the ratio $4:9$

Assume $ABC$ and $DEF$ are two similar triangles,

i.e $\Delta ABC~\Delta DEF$

so, $\frac{AB}{DE}=\frac{AC}{DF}=\frac{BC}{EF}=\frac{4}{9}$

Find the areas of triangles in the ratio:

Because the ratio of the areas of these triangles is equal to the square of the ratio of the adjacent angles,

$\begin{array}{rcl}\frac{\mathrm{Area}\left(\mathrm{\Delta ABC}\right)}{\mathrm{Area}\left(\mathrm{\Delta DEF}\right)}& =& \frac{{\mathrm{AB}}^2}{{\mathrm{DE}}^2}\\ & \Rightarrow & \frac{\mathrm{Area}\left(\mathrm{\Delta ABC}\right)}{\mathrm{Area}\left(\mathrm{\Delta DEF}\right)}={\left(\frac{4}{9}\right)}^2\\ & \Rightarrow & \frac{\mathrm{Area}\left(\mathrm{\Delta ABC}\right)}{\mathrm{Area}\left(\mathrm{\Delta DEF}\right)}=\frac{16}{81}\end{array}$

As a result, if the sides of two comparable triangles are in the ratio $4:9,$

So, the areas in the ratio is $16:81$ .

Explanation for the incorrect option:

The value of the ratio obtained above is not equivalent to the value in Option (A) Option (B) Option (C).

Hence, Option (A) Option (B) Option (C) are incorrect option.

Hence, option (D) is the correct answer