10. A rectangle is divided into three smaller congruent rectangles as shown.
Each smaller rectangle is similar to the large rectangle.
In each of the four rectangles, what is the ratio of the length of a longer side to that of a shorter side?

A $2 \sqrt{3}: 1$
B $3: 1$
C $2: 1$
D $\sqrt{3}: 1$
E $\sqrt{2}: 1$
Solution
D
We suppose that the length of the longer sides of the three smaller rectangles is $x$ and the length of their shorter sides is $y$.
It follows that the longer sides of the large rectangle have length $3 y$, and that its shorter sides have length $x$.
Because the smaller rectangles are similar to the larger rectangle $\frac{x}{y}=\frac{3 y}{x}$. Therefore $\frac{x^{2}}{y^{2}}=\frac{3}{1}$. Hence $\frac{x}{y}=\frac{\sqrt{3}}{1}$.
It follows that the ratio of the length of a longer side to that of a shorter side in all the rectangles is $\sqrt{3}: 1$.