To verify whether the diagonal distance across the rectangular soccer field is exactly 50 meters, we can use the Pythagorean Theorem. The theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

For a rectangle, the length and width form two sides of a right-angled triangle, and the diagonal is the hypotenuse.

Step-by-step process:

1. Identify the sides of the right-angled triangle:

   • One side is the length of the rectangle, which is 40 meters.
   • The other side is the width of the rectangle, which is 30 meters.
   • The diagonal is the hypotenuse of the triangle, which we need to verify.

2. Apply the Pythagorean Theorem: $c^2 = a^2 + b^2$ Where:

   • $c$ is the hypotenuse (diagonal),
   • $a$ is the length of the rectangle (40 meters),
   • $b$ is the width of the rectangle (30 meters).

3. Substitute the values: $c^2 = 40^2 + 30^2$ $c^2 = 1600 + 900$ $c^2 = 2500$

4. Solve for $c$ (the diagonal): $c = \sqrt{2500}$ $c = 50$

5. Conclusion: The diagonal distance across the field is indeed 50 meters, as the Pythagorean Theorem confirms this result.

So, the diagonal is exactly 50 meters.

Would you like a deeper explanation of the Pythagorean Theorem or how it applies to other scenarios?

Here are some additional questions related to this topic:

1. How would the diagonal change if the length of the soccer field were doubled but the width stayed the same?
2. Can the Pythagorean Theorem be used for non-right-angled triangles? Why or why not?
3. If the length was 50 meters and the width was 60 meters, what would the diagonal be?
4. How can the Pythagorean Theorem help in determining the shortest path across a rectangular area?
5. What happens if the lengths of the sides are negative? How does the Pythagorean Theorem handle that situation?

Tip: The Pythagorean Theorem is only valid for right-angled triangles, so it is important to ensure the angle between the two sides is 90 degrees before applying it.