Brilliant Geometry puzzle 》》 Rectangle puzzle

We note that the blue and yellow triangles are similar. Therefore that areas of the two triangles are proportional to the square of their linear dimensions. That is if the blue triangle has a base of $2a$ and height of $2h$ , the yellow triangle will have a base of $3a$ and height of $3h$ . Since $\dfrac {4ah}2 = 4$ and $\dfrac {9ah}2 = 9$ , $ah =2$ .

Let the area of the red region be $A$ . Then $A + 9 = \dfrac 12 (3a)(5h) = \dfrac {15ah}2 = 15 \implies A = 15-9 = \boxed 6$ .

blue triangle and yellow triangle are similar with ratio $\dfrac{2}{3}$ so (left side lenght of blue) / (right side lenght of yellow) = $\frac{2}{3}$ blue triangle and red one shares same height so area ratio will be $\dfrac{2}{3}$

$\dfrac{4}{S}=\dfrac{2}{3}$

$S=6$