Imagine we have two similar triangles ABC and DEF, and we need to find the ratio between the areas.
The formula for the area is $\frac{1}{2}hb$, where the $h$ is the height and $b$ is the base.
So in our case where we have bases $b$ and $e$ the ratio of the areas would be $\frac{\frac{1}{2}h_1b}{\frac{1}{2}h_2e}$
I see this formula shortened to just the ratio of their bases squared, so $\frac{b^2}{e^2}$.
First question: do we always put the smaller triangle in the numerator and the larger in the denominator?
Second question: why can the formula for the ratio between areas be shortened like that, is it because in the ratio between $h_1$ and $h_2$ is the same as $b$ and $e$ so we just swap the heights for bases?
Best Answer
By definition of similar triangles if Triangle 1 has Height $h_1$ and base $b_1$ and Triangle 2 has height $h_2$ and base $b_2$, then $h_2=kh_1$, $b_2=kb_1$ for some constant $k$.
$k=\frac{b_2}{b_1}=\frac{h_2}{h_1}$
Area 1 is $A_1=(1/2) h_1b_1$.
$A_2=(1/2)h_2b_2$
$\frac{A_2}{A_1}=(\frac{h_2}{h_1})(\frac{b_2}{b_1})=k^2=(\frac{b_2}{b_1})^2=(\frac{h_2}{h_1})^2$
Generally speaking, the ratio of the areas between two objects is the square of the ratio of their corresponding lengths.
If you consider $\frac{A_1}{A_2}=\frac{1}{k^2}$ you find the ratio between areas changes, but the principle still holds since $\frac{1}{k^2}=(\frac{b_1}{b_2})^2$