Question

Sides of two similar triangles are in 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

Hint: In this question we have to choose the correct option for the area of these triangles. The given problem is the two triangles are similar and are in ratio of \[4:9\]. By using some triangle properties we are going to find the ratio of these to triangles.
SSS (Side-Side-Side): If three pairs of sides of two triangles are equal in length, then the triangles are congruent. ASA (Angle-Side-Angle): If two pairs of angles of two triangles are equal in measurement, and the included sides are equal in length, then the triangles are congruent.

Complete step-by-step answer:
It is given that, Sides of two similar triangles are in ratio \[4:9\].

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.