Question

Ratio of corresponding sides of two similar triangles is 2:5, if the area of the smaller triangle is 64sq cm, then what is the area of the bigger triangle?

Answer 1

Hint: If two triangles are similar then we know the theorem “The ratio of the areas of two similar triangles is equal to the square of ratio of their corresponding sides”. Since the ratio is given and also the area of the smaller triangle is given, using the above theorem we can find the area of the bigger triangle.

Complete step-by-step answer:
If two triangles are said to be similar if they have the same shape, but can be of different size. That is bigger or smaller. \[ \Delta ABC \] and \[ \Delta PQR \] are similar.

Let area of \[ \Delta PQR = 64sq.cm \] . We need to find the area of \[ \Delta ABC \] .
“The ratio of the areas of two similar triangles is equal to the square of ratio of their corresponding sides”. That is \[ \dfrac{{{ \text{area of }} \Delta PQR}}{{{ \text{area of}} \Delta ABC}} = { \left( { \dfrac{{PQ}}{{AB}}} \right)^2} \] ----- (1)
But, the ratio between the corresponding sides of two triangles are given.
That is, \[ \dfrac{{PQ}}{{AB}} = \dfrac{2}{5} \] . Substituting this in (1).
 \[ \Rightarrow \dfrac{{64}}{{{ \text{area of}} \Delta ABC}} = { \left( { \dfrac{2}{5}} \right)^2} \]
 \[ \Rightarrow \dfrac{{64}}{{{ \text{area of}} \Delta ABC}} = \left( { \dfrac{4}{{25}}} \right) \]
Cross multiplying and rearranging the equation,
 \[ \Rightarrow \dfrac{{{ \text{area of}} \Delta ABC}}{{64}} = \dfrac{{25}}{4} \]
 \[ \Rightarrow { \text{area of}} \Delta ABC = \dfrac{{64 \times 25}}{4} \]
 \[ \Rightarrow { \text{area of}} \Delta ABC = \dfrac{{1600}}{4} \]
 \[ \Rightarrow { \text{area of}} \Delta ABC = 400sq.cm \]
Hence, the area of \[ \Delta ABC \] is 400 sq.cm
So, the correct answer is “400 sq.cm”.

Note: The theorem we mentioned in above is already proved. So using that theorem we can solve this type of question. For corresponding sides we took AB and PQ. You can also take AC and PR. (see the above diagram so that you can understand what we did in above). Similar triangles need not be equal in size, but shape will be the same in both (AB=5 and PQ=2)