$\Delta$ AOB and $\Delta$ DOC should be equal in area. Correct me if I am wrong.
Given: Trapezoid ABCD with ratio $\frac{area \Delta AOB}{area\Delta ABD}$ = $\frac{3}{4}$.
I am trying to find (1) Ratio of Area $\Delta$AOD to $\Delta$BOC; (2) Ratio of Area $\Delta$COD to Trapezoid ABCD
I know that the final solution (1) is 1:9 and (2) is 3:16, but I do not understand how to get to these answers.
Please explain how to get to the answer, and if possible, how might I find different ratios like $\Delta$BDC to trapezoid ABCD, or $\Delta$BDC to $\Delta$ABD?
Note: I have also seen this post but I'm not sure it's relevant to what I'm asking.
Best Answer
You can use the result of that post to conclude that the ratio is
$$\frac{a^2}{b^2}$$
The ratio is also
$$\frac{\frac{1}{2} a h_1}{\frac{1}{2} b h_2}$$
where $h_1,h_2$ are heights of $\Delta AOD$ and $\Delta BOC$, respectively. Now since
$$\frac{\text{area} \Delta AOB}{\text{area} \Delta ABD}=\frac{3}{4}$$
$$\frac{h_1}{h_2}=\frac{1}{3}$$
which gives you
$$\frac{a^2}{b^2}=\frac{a}{b} \cdot \frac{1}{3}$$
So $\frac{a}{b}=1/3$, that means the ratio of area of $\Delta AOD$ and $\Delta BOC$ is $1:9$.
I think you can then find the other one using similar argument.