Problem 5.8. Inside a large triangle with a perimeter of 120, several segments were drawn, dividing it into nine smaller triangles, as shown in the figure. It turned out that the perimeters of all nine small triangles are equal to each other. What can they be equal to? List all possible options.

The perimeter of a figure is the sum of the lengths of all its sides.

![](https://cdn.mathpix.com/cropped/2024_05_06_0323bbf84409a1adeb34g-12.jpg?height=256&width=323&top_left_y=1257&top_left_x=565)
Answer: 40.

Solution. Let's add the perimeters of the six small triangles marked in gray in the following figure:

![](https://cdn.mathpix.com/cropped/2024_05_06_0323bbf84409a1adeb34g-13.jpg?height=262&width=315&top_left_y=83&top_left_x=573)

From the obtained value, subtract the perimeters of the other three small white triangles. Since the perimeters of the small triangles are equal, on one side, we get three times the perimeter of a small triangle. On the other side, in this sum, there will only be segments that make up the perimeter of the large triangle, which is 120. Therefore, the perimeter of a small triangle is $120: 3=40$.

## 6th grade