Area inside a rectangle

Question

SEAMO 2016 paper E

I do not know how to start with this question and I have tried finding heights and bases of the triangle in terms of the sides of rectangle but i could not find the ideal pair

Answer 1

Let $AC\cap EG=\{H\}$ and $AC\cap GF=\{I\}$.

Thus, $$\frac{AH}{HC}=\frac{AE}{GC}=\frac{\frac{1}{3}}{\frac{1}{2}}=\frac{2}{3},$$ Which says that $$AH=\frac{2}{5}AC.$$ Also, $$\frac{CI}{IA}=\frac{CG}{AF}=\frac{\frac{1}{2}}{\frac{2}{3}}=\frac{3}{4},$$ Which says that $$CI=\frac{3}{7}AC.$$ Thus, $$HI=AC\left(1-\frac{3}{7}-\frac{2}{5}\right)=\frac{6}{35}AC,$$ which says $$\frac{S_{\Delta IHG}}{S_{\Delta ADC}}=\frac{6}{35}\cdot\frac{1}{2}=\frac{3}{35}$$ and $$S_{\Delta IHG}=\frac{3}{70}S_{ABCD}=1.5.$$

Answer 2

EFG is an isoceles triangle that has an area 1/6 the area of the whole rectangle

The green shaded area has the equivalent area to a triangle that is the top half of the triangle EFG. (Imagine a line horizontally between the midpoints of EG and EF, just as much green above as below the line)

The top half of an isoceles triangle has 1/4 the area of the whole

Therefore the area of the green shaded is 1/24 the are of the whole.

Assuming the total area is 36 then the shaded ares is 1.5

If the total area is 35 then maybe I got this wrong or its none of the above