"Imagine you’re designing a rectangular soccer field. You measure the length as 40 meters and the width as 30 meters. You want to check if the diagonal distance across the field is exactly 50 meters. How would you use the Pythagorean Theorem to verify this? Explain your thinking step by step."

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.

Imagine you’re designing a rectangular soccer field. You measure the length as 40 meters and the width as 30 meters. You want to check if the diagonal distance across the field is exactly 50 meters. How would you use the Pythagorean Theorem to verify this? Explain your thinking step by step.

Explore how to use the Pythagorean Theorem to verify the diagonal of a rectangular soccer field with dimensions 40m by 30m. This detailed step-by-step guide helps you understand the application of the theorem and confirm that the diagonal is 50 meters. Perfect for students in Grades 7-9.

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