Question

Given that \[\Delta DEF\sim \Delta ABC\], if \[DE:AB=2:3\] and \[\text{ar}\left( \Delta DEF \right)\] is equal to 44 square units. Find the area of \[\left( \Delta ABC \right)\].

Answer 1

Hint: In this type of question we have to use the concept of similarity of triangles. We know that two triangles are said to be similar if their corresponding angles are congruent and corresponding sides are in equal proportion. Also we know that if two triangles are similar to each other then the ratio of area of similar triangles is equal to the ratio of the square of the its corresponding sides.

Complete step by step answer:
Now, we have to find the area of \[\left( \Delta ABC \right)\] and given that \[\Delta DEF\sim \Delta ABC\], \[DE:AB=2:3\] and
\[\text{ar}\left( \Delta DEF \right)\]= 44 square units.

We know that, Ratio of area of similar triangles is equal to the ratio of the square of its corresponding sides.
\[\Rightarrow \dfrac{\text{ar}\left( \Delta DEF \right)}{\text{ar}\left( \Delta ABC \right)}={{\left( \dfrac{DE}{AB} \right)}^{2}}\]
Now substituting the values at their appropriate places we get,
\[\begin{align}
  & \Rightarrow \dfrac{44}{\text{ar}\left( \Delta ABC \right)}={{\left( \dfrac{2}{3} \right)}^{2}} \\
 & \Rightarrow \dfrac{44}{\text{ar}\left( \Delta ABC \right)}=\dfrac{4}{9} \\
\end{align}\]
Let us simplify it further so that we can write,
\[\begin{align}
  & \Rightarrow \dfrac{44\times 9}{4}=\text{ar}\left( \Delta ABC \right) \\
 & \Rightarrow 11\times 9=\text{ar}\left( \Delta ABC \right) \\
 & \Rightarrow \text{ar}\left( \Delta ABC \right)=99\text{ square units} \\
\end{align}\]
Hence, we get area of \[\left( \Delta ABC \right)\] is 99 square units.

Note: In this type of question students may make mistake when they consider the ratio. Students have to remember that the ratio of the area of similar triangles is equal to the ratio of the square of its corresponding sides. Here the word square is important as if it get missed student will get wrong answer. One of the students may write the same statement as “the ratio of area of similar triangles is equal to the square of the ratio of its corresponding sides”.