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?

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