# 5. Rectangle $A B C D$ was divided into four

 smaller rectangles with equal perimeters (see figure). It is known that $A B=18$ cm, and $B C=16$ cm. Find the lengths of the sides of the other rectangles. Be sure to explain your answer.
Answer. 2 cm and 18 cm are the lengths of the sides of rectangle $A B L E$, 6 cm and 14 cm are the lengths of the sides of the other rectangles.

Solution. Since the perimeters of the three vertical rectangles are equal and the segments $E D, F G, K H$ and $L C$ are also equal, the segments $E F$, $F K$ and $K L$ are also equal. Therefore, each of

![](https://cdn.mathpix.com/cropped/2024_05_06_843bb5844bda57b2e45dg-3.jpg?height=757&width=811&top_left_y=981&top_left_x=1185)
the segments $E F, F K$ and $K L$ is

$$
18 \text { cm : } 3=6 \text { cm. }
$$

The perimeter of rectangle $A B L E$ is equal to the perimeter of rectangle $D E F G$. We can solve this by setting up an equation or by trying to solve it without one.

Method 1. Let $A E=x$ cm, then $18+18+x+x=6+6+16-x+16-x$. From this, we find $x=2$. Therefore, $A E=2$ cm, and $E D=14$ cm.

Method 2. From the condition, it follows that $D E+E F=A E+A B$, that is, $D E+6=A E+18$. Then the difference in the lengths of segments $D E$ and $A E$ is 12 cm, and their sum is 16 cm according to the condition. Therefore, $A E=2$ cm, and $D E=14$ cm.

## Grading Criteria.

- Correct answer and complete explanation - 7 points.
- Correct answer with incomplete explanation - 5-6 points.
- Correct reasoning but with an arithmetic error - 3 points.
- Something is written on the picture, and the correct answer is obtained - 1-2 points.