Question

If the ratio of the perimeter of two similar triangles is \[9:16\]what the ratio of the area is?

Answer 1

Hint: In this question we are given the ratio of the perimeter of the triangle which is\[9:16\]. To solve this question we will use the perimeter property of similar triangles and as we are aware that the ratio of the areas of the two similar triangles is equal to the square of the ratio of the corresponding sides we will find the desired result.

Formula used:
Here we used the area property which states that the ratio of the areas of the two similar triangles is equal to the square of the ratio of the corresponding sides we will find the desired result.
So the formula used to find the ratio of the area will be \[{\left( {\dfrac{a}{b}} \right)^2}\]where \[a:b\]will be the ratio of the perimeter of two similar triangles.

Complete step-by-step answer:
Let the two triangles be ABC whose sides are a,b,c and PQR whose sides are p,q,r which are two similar triangles. Therefore \[\dfrac{a}{p} = \dfrac{b}{q} = \dfrac{c}{r} = k\]
Thus \[a = pk,{\text{ }}b = qk{\text{ }}and{\text{ }}c = rk\]
Now given that \[\dfrac{{(a + b + c)}}{{(p + q + r)}} = \dfrac{9}{{16}}\]. Here by putting \[a = pk,{\text{ }}b = qk{\text{ }}and{\text{ }}c = rk\] we get
\[k = \dfrac{9}{{16}}\]Where k is the ratio of the perimeter of two similar triangles
As we are aware that ratio of the areas of the two similar triangles is equal to the square of the ratio of the corresponding sides that is \[{(k)^2}\]
So ratio of the area is \[{\left( {\dfrac{9}{{16}}} \right)^2}\]and on further simplification we get
\[\left( {\dfrac{{81}}{{256}}} \right)\] or \[81:256\]

Note: Here the two triangles given are similar and as we know that just like the perimeter property we also have area property and this are property is used here to solve the above question which states that ratio of the areas of the two similar triangles is equal to the square of the ratio of the corresponding sides.