Problem 11 · The square cut in three
Equal areas pin the points; the distance 1 pins the side.
Integer answer, at most 4 digitsA square has been divided into three regions of equal area — two right triangles and a parallelogram — so that segments $AB$ and $CD$ are parallel and the distance between them is $1$ cm, as shown in the picture. What is the area of the square?
Copa Cangur · SCM
Hard
Closed answer
Reasoned solution
Key idea: place the square with $B = (0,0)$ and side $a$: the equal areas determine where $A$ and $D$ are.
The left triangle (vertices $B$, $(0,a)$ and $A = (x,a)$) has area $\tfrac{ax}{2} = \tfrac{a^2}{3}$, so $x = \tfrac{2a}{3}$. By symmetry, $D = \left(\tfrac{a}{3}, 0\right)$.
Line $BA$ is $3x - 2y = 0$ and line $DC$ is $3x - 2y = a$. The distance between them:
$$d = \frac{a}{\sqrt{3^2 + 2^2}} = \frac{a}{\sqrt{13}} = 1 \;\Longrightarrow\; a^2 = 13\ \text{cm}^2.$$
Answer: 13