Side of two similar triangles are in the ratio 4:9 area of these triangles are in the ratio

Correct Answer: (d) 16:81

Explanation:

If two triangles are similar, then the ratio of their areas is equal to the ratio of the squares of their corresponding sides.

$\begin{array}{l}

\frac{{area\ of\ first\ triangle}}{{area\ of\ \second\ triangle}} = {\left( {\frac{{{side\ of\ first\ triangle/}}{{side\ of\ \second\ triangle}}} \right)^2}\\

= > {\left( {\frac{4}{9}} \right)^2}\\

= > \frac{{16}}{{81}}

\end{array}$

Answer

Verified

Let, \[\Delta ABC\] and \[\Delta DEF\] are the two given similar triangles. We need to find the ratio of \[{{{\text{Area}}(\Delta ABC)}}:{{{\text{Area}}(\Delta DEF)}}\].Thus we have their sides in the ratio \[4:9\] .\[ \Rightarrow {{AB}}:{{DE}} = {{AC}}:{{DF}} = {{BC}}:{{EF}} = {4}:{9}\; \ldots .\left( 1 \right)\]We know that if two triangle are similar,Ratio of areas is equal to square of ratio of its corresponding sides\[ \Rightarrow {{{\text{Area}}(\Delta ABC)}}:{{{\text{Area}}(\Delta DEF)}} = {\left( {{{BC}}:{{EF}}} \right)^2}\] Putting the values in (1)\[ \Rightarrow {{{\text{Area}}(\vartriangle ABC)}}:{{{\text{Area}}(\vartriangle DEF)}} = {\left( {{4}:{9}} \right)^2} = {{16}}:{{81}}\]Hence, the areas of triangles \[\Delta ABC\] and \[\Delta DEF\] is: \[16:81\] .Hence, when sides of two similar triangles are in ratio \[4:9\]. Areas of these triangles are in the ratio: \[16:81\].

(D) is the correct option.

Note: If an angle of one triangle is congruent to the corresponding angle of another triangle and the lengths of the sides including these angles are in proportion, the triangles are similar. The corresponding sides of similar triangles are in proportion.

We have used the following theorem.Theorem: If two triangles are similar, then the ratio of the area of both triangles is proportional to the square of the ratio of their corresponding sides.

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

If two triangles are similar to each other, then the ratio of the areas of these triangles will be equal to the square of the ratio of the corresponding sides of these triangles.

It is given that the sides are in the ratio 4:9.

Therefore, ratio between areas of these triangles = `(4/9)^2 = 16/81`

Hence, the correct answer is 16 : 81.

Concept: Areas of Similar Triangles

Is there an error in this question or solution?

Solution:

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides

Given that,

Sides of two similar triangles are in the ratio 4: 9.

We know that,

The ratio of the areas of two similar triangles = square of the ratio of their corresponding sides

= (4: 9)2