Find unknown area of the triangle

Question

Find the area of the triangle. As I understand, if we name the unkown area x, then x/120=300/200 or x/300=120/200 and from that x=180, but I have no idea how to prove that it's correct.

Answer 1

One good way to more easily visualize this sort of solution is by adding to each triangle another, congruent triangle to make each a parallelogram, as in this very bad MS paint drawing:

From here I find it easier to see why

$$\frac{2?}{2\cdot 120} = \frac{2\cdot 300}{2\cdot 200},$$

even though it is true for the exact same reason (base times height).

Answer 2

Referring to the above drawing, a few details:

Let

1)$ x$ be the common basis of the $2×?$ and $2×120 $ parallelograms.

2) $y$ be the common basis of the $2×300$ and $2×200$ parallelograms.

3) $H$ the height of the $2×?$ parallelogram.

4) $h$ the height of the $2×120$ parallelogram.

We have:

a) $yH=600;$

b)$yh= 400$;

c) $xh= 240. $

Find: $xH.$

Divide a) by b): $H/h = 600/(400).$

Multiply this by c):

$xH = \dfrac{240×600}{400} = 2×?.$