7. Ana started drawing a rectangle $A B C D$ starting from point $A$. She drew the segments $\overline{A B}, \overline{B C}$, and $\overline{C D}$. While drawing the segment $\overline{D A}$, she reached point $E$ and instead of drawing segment $\overline{E A}$, she drew seven segments in a row and thus reached point $A$, as shown in her drawing. Each segment she drew is perpendicular to the previously drawn segment.

In the drawing, she marked the lengths of two segments, $|B C|=40 \mathrm{~cm}$ and $|C D|=5 \mathrm{~cm}$. What is the length of the last drawn segment $\overline{F A}$ if the perimeter of the drawn shape is $140 \mathrm{~cm}$?

![](https://cdn.mathpix.com/cropped/2024_06_03_6e1b38aefafca10e6d1eg-2.jpg?height=708&width=578&top_left_y=608&top_left_x=1273)

## Ministry of Science and Education of the Republic of Croatia  Agency for Education and Upbringing  Croatian Mathematical Society

## SCHOOL/CITY COMPETITION IN MATHEMATICS  January 26, 2024.
7. Ana started drawing a rectangle $A B C D$ starting from point $A$. She drew the segments $\overline{A B}, \overline{B C}$, and $\overline{C D}$. While drawing the segment $\overline{D A}$, she reached point $E$ and instead of drawing segment $\overline{E A}$, she drew seven segments in a row and thus reached point $A$, as shown in her drawing. Each segment she drew is perpendicular to the previously drawn segment.

In the drawing, she marked the lengths of two segments, $|B C|=40 \mathrm{~cm}$ and $|C D|=5 \mathrm{~cm}$. What is the length of the last drawn segment $\overline{F A}$ if the perimeter of the drawn shape is $140 \mathrm{~cm}$?

## Solution.

![](https://cdn.mathpix.com/cropped/2024_05_30_0b00c6f52e60bdba44a4g-06.jpg?height=623&width=505&top_left_y=251&top_left_x=1301)

Let $a$ be the length of the last drawn segment, $\overline{F A}$.

The quadrilateral $A B C D$ is a rectangle, so $|A B|=5 \mathrm{~cm}$.

1 POINT

Notice that the total length of the four segments that Ana drew parallel to segment $\overline{B C}$ is $40 \mathrm{~cm}$.

Similarly, the sum of the lengths of the three segments she drew after point $E$, which are parallel to segment $\overline{F A}$, is equal to the length of segment $\overline{F A}$, which is $a$.

The perimeter of the drawn shape is $140 \mathrm{~cm}$ and is equal to the length of the boundary of that shape, i.e., the sum of the lengths of all the segments she drew.

![](https://cdn.mathpix.com/cropped/2024_05_30_0b00c6f52e60bdba44a4g-06.jpg?height=195&width=1543&top_left_y=1453&top_left_x=248)

From this, it follows that the two lengths $a$ are equal to $140-5-5-40-40=50 \mathrm{~cm}$.

Therefore, the length $a$ of the desired segment is equal to $50: 2=25 \mathrm{~cm}$.

Note: Instead of the argument about the segments that are parallel to the sides of the rectangle $A B C D$, a diagram can also be drawn and it can be emphasized that the resulting drawn shapes are rectangles, which is worth 5 points. Based on this diagram, the expression for the perimeter of the shape that Ana drew is written.

![](https://cdn.mathpix.com/cropped/2024_05_30_0b00c6f52e60bdba44a4g-06.jpg?height=508&width=414&top_left_y=1893&top_left_x=1409)

# SCHOOL COMPETITION IN MATHEMATICS