Triangle Area Ratio Theorem Problems?

Question

Having a hell of a lot of issues with these problems, supposed to be on the topic of triangle area ratio theorem (ratio area of triangles = ratio of triangles' heights x ratio of triangles' bases.)

Here are that are causing me trouble:

Any help would be greatly appreciated.

Answer 1

I'll get you started:

For (2), we use $\text{Area} = \frac{1}{2}ab\sin(C)$, where $C$ is the angle between the sides with length $a$ and $b$. So, $$ [CDE] = \frac{1}{2} \times 3 \times 5 \times \sin(\angle ACB) = 7, $$ Use this to find $\sin(\angle ACB)$, from which you can use the same formula to find $[ABC]$.

For (3), the same formula can get you $\angle ACB$, from which you can determine everything about $\triangle ABC$ - in particular, the information you need to find $x$.