Problem 3 · Two grey squares under the diagonal
Similarity along the diagonal of a 2×1 rectangle and an area percentage.
Integer answer, at most 4 digitsWe build a rectangle out of two squares and draw its diagonal. Two grey squares are placed as shown in the figure. What percentage of the original rectangle is occupied by the two grey squares?
Copa Cangur · SCM
Medium
Closed answer
Reasoned solution
Key idea: choose convenient measurements (the result is a percentage, so it is scale-free): a $2 \times 1$ rectangle with diagonal from $(0,0)$ to $(2,1)$, i.e. the line $y = x/2$.
The two grey squares, of side $s$, are stacked in the bottom-right corner: they occupy $[2-s,\,2] \times [0,\,2s]$. The top-left corner of the stack, $(2-s,\; 2s)$, lies on the diagonal:
$$2s = \frac{2-s}{2} \;\Longrightarrow\; 4s = 2 - s \;\Longrightarrow\; s = \frac{2}{5}.$$
Grey area: $2s^{2} = 2\cdot\dfrac{4}{25} = \dfrac{8}{25}$. Rectangle area: $2$. Percentage:
$$\frac{8/25}{2} = \frac{4}{25} = 16\,\%.$$
Answer: 16