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Hint: The dimensional proportionality of similar triangles claims the ratio of areas of triangles to be the square of the ratio of their corresponding sides.

Here we are given that the area of two similar triangles are in the ratio 9:16

Let us consider that

A1 and A2 are the areas of two similar triangle, and

S1 and S2 are the corresponding sides of two similar triangles.

THEOREM:

The ratio of square of corresponding side of two similar triangles is equal to ratio of area of two similar triangles.

Now by using the above theorem the ratio of area of two similar triangles having areas A1 and A2 can be written as

$\dfrac{{A1}}{{A2}} = {\left( {\dfrac{{S1}}{{S2}}} \right)^2}$

Here the ratio of area of two similar triangles is given as 9:16

On substituting the values, we get

$\dfrac{9}{{16}} = {\left( {\dfrac{{S1}}{{S2}}} \right)^2}$

Taking square root on both sides, we get

$\dfrac{{S1}}{{S2}} = \dfrac{3}{4}$

Therefore the ratio of corresponding sides of similar triangle will be 3:4

Note: Here we have used the dimensional proportionality of similar triangles. This says that the ratio of square of corresponding side of two similar triangles is equal to ratio of area of two similar triangles. Now on substituting values we get the ratio of corresponding sides of similar triangles. Make a note that we have to choose proper dimensional proportionality based on the question that we have found as there are any statements.

Here we are given that the area of two similar triangles are in the ratio 9:16

Let us consider that

A1 and A2 are the areas of two similar triangle, and

S1 and S2 are the corresponding sides of two similar triangles.

THEOREM:

The ratio of square of corresponding side of two similar triangles is equal to ratio of area of two similar triangles.

Now by using the above theorem the ratio of area of two similar triangles having areas A1 and A2 can be written as

$\dfrac{{A1}}{{A2}} = {\left( {\dfrac{{S1}}{{S2}}} \right)^2}$

Here the ratio of area of two similar triangles is given as 9:16

On substituting the values, we get

$\dfrac{9}{{16}} = {\left( {\dfrac{{S1}}{{S2}}} \right)^2}$

Taking square root on both sides, we get

$\dfrac{{S1}}{{S2}} = \dfrac{3}{4}$

Therefore the ratio of corresponding sides of similar triangle will be 3:4

Note: Here we have used the dimensional proportionality of similar triangles. This says that the ratio of square of corresponding side of two similar triangles is equal to ratio of area of two similar triangles. Now on substituting values we get the ratio of corresponding sides of similar triangles. Make a note that we have to choose proper dimensional proportionality based on the question that we have found as there are any statements.