1. Francisca has a square piece of paper with sides of $10 \mathrm{~cm}$. She also has a rectangular piece of paper with exactly the same area as the square piece of paper. She neatly places the rectangle on top of the square, so that the corners at the bottom left of both sheets coincide exactly. Exactly one quarter of the square remains uncovered by the rectangle.

How many centimeters long is the longest side of the rectangle?
A) 12
B) $12 \frac{1}{4}$
C) $12 \frac{1}{2}$
D) $12 \frac{3}{4}$
E) $13 \frac{1}{3}$

![](https://cdn.mathpix.com/cropped/2024_04_17_cc2ba9e3b1603277d398g-1.jpg?height=326&width=260&top_left_y=1256&top_left_x=1549)
A1. E) $13 \frac{1}{3} \quad$ The right side of the rectangle intersects the top and bottom sides of the square at exactly three-quarters of those sides. This means that $A B=7 \frac{1}{2} \mathrm{~cm}$. The rectangle has the same area as the square: $10 \mathrm{~cm} \times 10 \mathrm{~cm}=100 \mathrm{~cm}^{2}$. Therefore, the length of the long side of the rectangle is

$$
\frac{100 \mathrm{~cm}^{2}}{7 \frac{1}{2} \mathrm{~cm}}=\frac{200}{15} \mathrm{~cm}=\frac{40}{3} \mathrm{~cm}=13 \frac{1}{3} \mathrm{~cm}
$$

![](https://cdn.mathpix.com/cropped/2024_04_17_2ddd9533af815a0f4acbg-1.jpg?height=368&width=263&top_left_y=593&top_left_x=1522)