To determine the length of one side of a square playground given the length of the diagonal walkway, we need to use the Pythagorean theorem. The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (diagonal of the square in this case) is equal to the sum of the squares of the lengths of the other two sides (which are the sides of the square).
Let’s denote:
- [tex]\( a \)[/tex] as the length of one side of the square.
- [tex]\( c \)[/tex] as the length of the diagonal walkway, which is 55 meters.
In a square, the diagonal splits the square into two right-angled triangles. Therefore, the diagonal ([tex]\( c \)[/tex]) can be related to the side of the square ([tex]\( a \)[/tex]) using the Pythagorean theorem:
[tex]\[
a^2 + a^2 = c^2
\][/tex]
Since [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are the same in a square, we can simplify this to:
[tex]\[
2a^2 = c^2
\][/tex]
We can solve for [tex]\( a \)[/tex]:
[tex]\[
a^2 = \frac{c^2}{2}
\][/tex]
Now, let's substitute [tex]\( c = 55 \)[/tex]:
[tex]\[
a^2 = \frac{55^2}{2}
\][/tex]
[tex]\[
a^2 = \frac{3025}{2}
\][/tex]
[tex]\[
a^2 = 1512.5
\][/tex]
Taking the square root of both sides to solve for [tex]\( a \)[/tex]:
[tex]\[
a = \sqrt{1512.5}
\][/tex]
Using a calculator, we get:
[tex]\[
a \approx 38.890872965260115
\][/tex]
To round this to the nearest tenth:
[tex]\[
a \approx 38.9
\][/tex]
So, the length of one side of the playground, rounded to the nearest tenth, is 38.9 meters.
Here is a sketch of the situation:
```
+-----------------+
| |
| |
| |
| |
| di |
| a g |
| onal |
| (55m) |
| |
| |
| |
| |
+-----------------+
a
```
Therefore, the length of one side of the square playground is approximately 38.9 meters.
