5) Knowing that the rectangle in the figure is divided by the inclined line into two parts, one of which has an area four times that of the other, calculate the ratio between the measures of segments $a$ and $b$.
(A) $2 / 3$
(B) $1 / 4$
(C) $1 / 5$
(D) $1 / 2$
(E) $2 / 5$

![](https://cdn.mathpix.com/cropped/2024_04_17_62915d10d7246845095ag-1.jpg?height=216&width=298&top_left_y=310&top_left_x=2443)
5. The answer is $(\mathbf{A})$.

Let $X$ and $Y$ be the measures (in a given unit of measurement) of the base and the height of the rectangle, respectively, and let $A$ and $B$ be the respective measures of segments $a$ and $b$. We know that $A+B=Y$ and

$$
\frac{\frac{1}{2}(B+Y) \cdot X}{\frac{1}{2} A \cdot X}=4
$$

and therefore

$$
4=\frac{A+2 B}{A}=1+2 \frac{B}{A} \Rightarrow \frac{B}{A}=\frac{3}{2}
$$

or $\frac{A}{B}=\frac{2}{3}$

[Problem proposed by F. Mugelli]