4. Lea drew a rectangle where one side is longer than the other by $3 \mathrm{~cm}$. After that, she drew two equilateral triangles on each of the shorter sides of the rectangle such that their sides are of equal length, as shown in the figure. If the perimeter of the resulting shape is $78 \mathrm{~cm}$, what is the perimeter of the rectangle?

![](https://cdn.mathpix.com/cropped/2024_06_03_1d05bbb73aad96dd359ag-1.jpg?height=379&width=742&top_left_y=1815&top_left_x=1111)
4. Lea drew a rectangle where one side is longer than the other by $3 \mathrm{~cm}$. After that, she drew two equilateral triangles on each of the shorter sides of the rectangle such that their sides are equally long, as shown in the figure. If the perimeter of the resulting shape is $78 \mathrm{~cm}$, what is the perimeter of the rectangle?

![](https://cdn.mathpix.com/cropped/2024_05_30_2e894aa57f8bdfbd97cfg-06.jpg?height=323&width=629&top_left_y=227&top_left_x=1179)

## Solution.

Let the side length of the equilateral triangle be $a$ in centimeters.

Then the length of the shorter side of the rectangle is $a+a=2a$.

So, the length of the longer side of the rectangle is $2a+3$.

3 POINTS

The perimeter of the shape in the figure is equal to the length of its boundary, that is,

![](https://cdn.mathpix.com/cropped/2024_05_30_2e894aa57f8bdfbd97cfg-06.jpg?height=411&width=711&top_left_y=591&top_left_x=1141)

$2a+3$ is the sum of the lengths of its sides.

The eight segments that bound the shape are sides of the equilateral triangles and their length is $a$. The remaining two segments that bound the shape are the longer sides of the rectangle and their length is $2a+3$. The boundary of this shape can be arranged into a line segment as follows:

![](https://cdn.mathpix.com/cropped/2024_05_30_2e894aa57f8bdfbd97cfg-06.jpg?height=71&width=1356&top_left_y=1341&top_left_x=356)

This means that the perimeter of this shape is $12a+6$.

2 POINTS

Now we have:

$12a+6=78$

$12a=72$

$a=6 \mathrm{~cm}$.

2 POINTS

The shorter side of the rectangle has a length of $6+6=12 \mathrm{~cm}$, and the longer side is $12+3=15 \mathrm{~cm}$.

1 POINT

The perimeter of the rectangle is $2 \cdot (12+15) = 2 \cdot 27 = 54 \mathrm{~cm}$.

2 POINTS

TOTAL 10 POINTS