To verify if the diagonal distance of the rectangular soccer field is exactly 50 meters using the Pythagorean Theorem, we can proceed step by step. Here's the process:

Step 1: Understand the Pythagorean Theorem

The Pythagorean Theorem states that in a right-angled triangle, the square of the length of the hypotenuse ($c$) is equal to the sum of the squares of the lengths of the other two sides ($a$ and $b$):

$c^2 = a^2 + b^2$

In the case of the rectangular soccer field, the length and width of the field are the two sides of the right-angled triangle, and the diagonal is the hypotenuse.

Step 2: Label the sides of the rectangle

From the problem, we know the following:

• Length of the field = 40 meters ($a = 40$)
• Width of the field = 30 meters ($b = 30$)

You want to find the length of the diagonal ($c$), and check if it’s 50 meters.

Step 3: Apply the Pythagorean Theorem

We will plug the values of the length and width into the Pythagorean Theorem to solve for the diagonal $c$.

$c^2 = a^2 + b^2$

$c^2 = 40^2 + 30^2$

$c^2 = 1600 + 900$

$c^2 = 2500$

Now, take the square root of both sides to find $c$:

$c = \sqrt{2500}$

$c = 50$

Step 4: Conclusion

The diagonal distance across the field is exactly 50 meters, which confirms that the measurement you have is correct!

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Would you like further details on how to apply this theorem to other shapes or scenarios? Let me know if you have more questions!

Related questions:

1. How does the Pythagorean Theorem apply to different shapes, like triangles or parallelograms?
2. How can the Pythagorean Theorem be used in real-world applications beyond geometry?
3. Can we use the Pythagorean Theorem to find the diagonal of a rectangular prism (3D)?
4. What happens if the rectangle is not a right-angled triangle? Can the theorem still be applied?
5. What are some alternative methods to calculate the diagonal of a rectangle?

Tip: The Pythagorean Theorem is most commonly used for right-angled triangles, but it also helps in situations like determining the shortest distance between two points on a grid!