# The ratio of the areas of two similar triangles is 1:k. What is the ratio of the lengths of their corresponding sides in terms of k?

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justaguide | Certified Educator

The two triangles are similar. If the sides of one of the triangles is a, b and c, the sides of the other triangle are a*f, b*f and c*f with f being a constant.

The formula of the area of a triangle with sides a, b, c is given by:

sqrt[s(s - a)(s - b)(s - c)] where s = (a + b + c)/2

As the areas of the two triangles are in the ratio 1:k, we get:

sqrt[s(s - a)(s - b)(s - c)] / sqrt[f*s(f*s - f*a)(f*s - f*b)(f*s - f*a) = 1/k

=> sqrt[s(s - a)(s - b)(s - c)] / (sqrt f)*sqrt[s(s - a)(s - b)(s - a) = 1/k

=> 1/sqrt f = 1/k

=> sqrt f = k

=> f = k^2

**This gives the ratio of the lengths of the sides as 1/k^2**