Problem 11 · The grey squares under the diagonal
Similarity on the 2×1 rectangle's diagonal: grey side $= \tfrac{2}{5}$ of the white one.
Integer answer, at most 4 digitsThe figure shows two white squares placed side by side, and two smaller grey squares placed inside the right-hand white square. If the area of a white square is 200 cm², what is the area of a grey square?
Copa Cangur · SCM
Medium
Closed answer
Reasoned solution
Key idea: the two white squares form a $2s \times s$ rectangle with its corner-to-corner diagonal drawn: the line $y = x/2$.
The two grey squares, of side $t$, are stacked in the bottom-right corner, and the top-left corner of the stack, $(2s - t,\ 2t)$, lies on the diagonal:
$$2t = \frac{2s - t}{2} \;\Longrightarrow\; 4t = 2s - t \;\Longrightarrow\; t = \frac{2s}{5}.$$
The area of one grey square:
$$t^{2} = \frac{4s^{2}}{25} = \frac{4 \cdot 200}{25} = 32\ \text{cm}^{2}.$$
Answer: 32