Question

The ratio of the areas of two similar triangles is $25:16$ . The ratio of their perimeters is $..........$ .
$\left( a \right){\text{ 625:256}}$
$\left( b \right){\text{ 5:6}}$
$\left( c \right){\text{ 25:16}}$
$\left( d \right){\text{ 5:4}}$

Answer 1

Hint: Since we have the ratio of the area of a triangle given. And as we realize that the proportion of regions of two triangles will be the same as the proportion of squares of their relating sides. And by using this statement we will be able to answer this question.

Complete step-by-step answer:
In this question first of all we will see the ratio given to us. So we have the ratio of the area of a triangle is $25:16$ . And as we realize that the proportion of regions of two triangles will be the same as the proportion of squares of their relating sides.
Therefore, the ratio of the corresponding sides of a similar triangle is $\sqrt {\dfrac{{25}}{{16}}} $ .
And on solving the under root, we get it as
$ \Rightarrow \dfrac{5}{4}$
And in the ratio, it can be termed as $5:4$
Hence, the ratio of their perimeter will be equal to $5:4$ .
Therefore, the option $\left( d \right)$ is correct.

Note: This question can also be solved by supposing the sides of both the triangle as ${a_1},{b_1},{c_1}$ and for the other it will be as ${a_2},{b_2},{c_2}$ . Therefore the perimeter will be named as ${P_1}$ and ${P_2}$ . And as we know the perimeter is calculated by summing up the sides of it. So from here we will have the ratios of the sides of a triangle and in the last, we will find the ratio of the perimeter and it will be equal to $\dfrac{{{P_1}}}{{{P_2}}} = \dfrac{{{a_1} + {a_2} + {a_3}}}{{{b_1} + {b_2} + {b_3}}} = \dfrac{5}{4}$ . And in this way, we can solve this question. Also while solving this type of question we should always mention the terms you are using that solution.