Question

The areas of similar triangles `DeltaABC` and `DeltaDEF` are equal. If the longest side of `DeltaDEF` measures 25 units, what is the length of the longest side of `DeltaABC`?

Answer 1

If two triangles are similar, it means that one is the scaled version of the other.

In similar triangles, correspondant angles are the same, and correspondant lengts are all in the same ratio $r$. The ratio between the areas is $r^2$.

So, for example, if the scale ratio is $r = 2$, all sides of the new triangle are twice as long as the sides of the old one, and the area of the new one is $r^2 = 4$ times the area of the old one.

Since in this case the two areas are equal, their ratio is $r^2 = 1$. This implied that also $r = 1$, which means that also correspondant sides are equal.

In other words, these two triangles are not only similar, but congruent. This means that, in particular, all correspondant sides have same length, so the longest side of ABC has a length of 25 as well.