https://file.aiquickdraw.com/mathbot/file/17301100343321t0kpesp.jpg

The problem shown in the image is centered around the **Pythagorean Theorem**, as suggested by the squares built on the sides of the triangle.

The image asks about the relationship between the lengths of sides \( AB \), \( BC \), and \( AC \) based on the areas of the squares on those sides.

Here is how we approach the problem:

1. **Step 1: Labeling the Sides**  
   The triangle \( ABC \) is a right triangle with:
   - \( AB \) as one leg
   - \( BC \) as the other leg
   - \( AC \) as the hypotenuse

2. **Step 2: Area of the Squares**
   - The area of the square on side \( AB \) is given by \( AB^2 \), which is the side length \( AB \times AB \).
   - The area of the square on side \( BC \) is \( BC^2 \), which is \( BC \times BC \).
   - The area of the square on side \( AC \) is \( AC^2 \), which is \( AC \times AC \).

3. **Step 3: Applying the Pythagorean Theorem**
   The Pythagorean Theorem states that for a right triangle:
   \[
   AC^2 = AB^2 + BC^2
   \]
   This formula shows that the area of the square on the hypotenuse \( AC \) is equal to the sum of the areas of the squares on the legs \( AB \) and \( BC \).

4. **Conclusion**  
   The relation between the side lengths is described by:
   \[
   AC^2 = AB^2 + BC^2
   \]

If you want, I can now go through the specific areas as per the provided numbers in the squares, or answer any further questions.

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Would you like to calculate the actual areas from the grid, or clarify any step?

Here are 5 follow-up questions:
1. How can we calculate the areas of the squares for each side?
2. What is the significance of the Pythagorean theorem in geometry?
3. Can we apply this concept to any right triangle?
4. How do the areas visually confirm the Pythagorean theorem?
5. What happens if the triangle is not a right triangle?

**Tip**: The Pythagorean theorem is a powerful tool for calculating distances in many geometric applications!

Based on the area of squares with unit sides, what is the relationship between the lengths AB, BC, and AC?

Explore the relationship between the lengths AB, BC, and AC in a right triangle using the Pythagorean theorem. Learn how the areas of squares built on triangle sides demonstrate this fundamental concept in geometry. Suitable for Grade 8-10 students.

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