Question

If $\mathrm{\Delta }ABC\sim DEF$ , $AB:DE=3:4$ , then find the ratio of area of triangles.

Answer 1

Hint: It is given that the triangle ABC is similar to the triangle DEF. This means that the two triangles have the shape but are different in size. In order to answer this question, we have to first understand the properties of similar triangles.

Complete step-by-step answer:
Given to us are two triangles ABC and DEF that are similar to each other. Hence the triangles ABC and DEF have the same shape and only differ in their size. This means that their sides should be in some ratio.
Here, the two sides of the triangles are in ratio $AB:DE=3:4$
According to properties of similar triangles
a) Two triangles are said to be similar if their angles are congruent i.e. same.
b) All the corresponding sides are in proportion.
So, if two triangles are similar then the ratio of their area will be the square of the ratio of one of their sides.
We can write this as
 ${\displaystyle \frac{A\left(\mathrm{\Delta }ABC\right)}{A\left(\mathrm{\Delta }DEF\right)}}={\left({\displaystyle \frac{AB}{DE}}\right)}^2$
Let us now substitute the given ratio of the sides in the above equation.
We get
 ${\displaystyle \frac{A\left(\mathrm{\Delta }ABC\right)}{A\left(\mathrm{\Delta }DEF\right)}}={\left({\displaystyle \frac{3}{4}}\right)}^2$
On solving this equation, we get ${\displaystyle \frac{A\left(\mathrm{\Delta }ABC\right)}{A\left(\mathrm{\Delta }DEF\right)}}={\displaystyle \frac{9}{16}}$
Hence, we can conclude that the ratio of areas of the two similar triangles given is ${\displaystyle \frac{9}{16}}$
So, the correct answer is “ ${\displaystyle \frac{9}{16}}$ ”.

Note: It should be noted that two similar triangles can share their sides or vertices. This means that two triangles could be on the same base or are a part of a bigger triangle, as long as they hold the same shape and congruent angle they are said to be similar.