Question

Sides of two simlar triangle are in ratio 4:9 area of these triangl in ratio

Answer 1

Question: Sides of two similar triangles are in the ratio 4:9. Find the ration of their areas.

Solution:

Let us consider two triangles ABC and PQR, who are similar and the sides are in the ratio 4:9.

We know that the ratio of the area of similar triangles is equal to the ratio of the square of their sides.

$\large\boxed{\sf \dfrac{ar(ABC)}{ar(PQR)} = \huge \text[\dfrac{AB}{PQ} \huge \text]^{2} =\huge \text[\dfrac{BC}{QR} \huge \text]^{2} =\huge \text[\dfrac{AC}{PR} \huge \text]^{2}}$

Now,

$\implies \sf \dfrac{ar(ABC)}{ar(PQR)} = \huge \text[\dfrac{AB}{PQ} \huge \text]^{2}$

$\implies \sf \dfrac{ar(ABC)}{ar(PQR)} = \huge \text[\dfrac{4}{9} \huge \text]^{2}$

$\implies \sf \dfrac{ar(ABC)}{ar(PQR)} = \dfrac{16}{81}$

$\large\boxed{\implies \sf ar(ABC):ar(PQR) = 16:81}$

$\large \text{\sf \underline {ANSWER}}$

16:81

Answer 2

Given :

• ∆ABC $\sim$ ∆PQR
• ratio of their sides = 4:9

To find :

• Ratio of their areas

Solution :

Let PQ be 4k and AB be 9k.

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

We know that, if two triangles are similar, then ratio of their areas is equal to the square of ratio of their corresponding sides.

⠀⠀⠀⠀⠀⠀⠀⠀⠀

$\therefore$ $\sf{\dfrac{area\:\triangle ABC}{area\:\triangle PQR}\:=\:\dfrac{AB^2}{PQ^2}}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀

Putting the values,

⠀⠀⠀⠀⠀⠀⠀⠀⠀

$\implies$ $\sf{\dfrac{Area\:of\:\triangle\:ABC}{Area\:of\:\triangle PQR}\:=\:\dfrac{(4k)^2}{(9k)^2}}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀

: $\implies$ $\sf{\dfrac{Area\:of\:\triangle\:ABC}{Area\:of\:\triangle PQR}\:=\:\dfrac{16k^2}{81k^2}}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀

Cancellation k² from numerator and denominator.

⠀⠀⠀⠀⠀⠀⠀⠀⠀

$\therefore$ Ratio of their areas is 16:81.