[Math] Given a ratio of the height of two similar triangles and the area of the larger triangle, calculate the area of the smaller triangle

Question

Please help, I've been working on this problem for ages and I can't seem to get the answer.

The heights of two similar triangles are in the ratio 2:5. If the area of the larger triangle is 400 square units, what is the area of the smaller triangle?

Answer 1

Hint: When you scale the linear dimensions of a figure by the factor $\lambda$, the area gets scaled by the factor $\lambda^2$. To get from the big triangle to the small one, you scale linear dimensions by the factor $\lambda=\frac{2}{5}$.

Or, in a version I like much less, take a triangle with base $b$, height with respect to that base $h$. Then new base, new height are $\lambda b$, $\lambda h$. So old area is $\frac{1}{2}bh$, new area is $\frac{1}{2}(\lambda b)(\lambda h)=\lambda^2 \frac{1}{2}bh$.