Jerry drew a figure on the board claiming that it was a right triangle. What side lengths of the figure would show that Jerry is correct?

To determine if the figure Jerry drew is a right triangle, we need to verify whether the side lengths of the triangle satisfy the Pythagorean theorem. For a triangle with side lengths \(a\), \(b\), and \(c\) (where \(c\) is the longest side, also known as the hypotenuse), the Pythagorean theorem states:

\[
a^2 + b^2 = c^2
\]

### Example of Side Lengths for a Right Triangle

To show that Jerry's triangle is indeed a right triangle, its side lengths should follow the equation above. Here are some examples of side lengths that satisfy this condition:

1. **3, 4, 5**  
   \[
   3^2 + 4^2 = 9 + 16 = 25 = 5^2
   \]
   So, \((3, 4, 5)\) is a set of side lengths for a right triangle.

2. **5, 12, 13**  
   \[
   5^2 + 12^2 = 25 + 144 = 169 = 13^2
   \]
   So, \((5, 12, 13)\) is another set of side lengths for a right triangle.

3. **7, 24, 25**  
   \[
   7^2 + 24^2 = 49 + 576 = 625 = 25^2
   \]
   So, \((7, 24, 25)\) is another example.

4. **8, 15, 17**  
   \[
   8^2 + 15^2 = 64 + 225 = 289 = 17^2
   \]
   So, \((8, 15, 17)\) is another set of side lengths.

5. **9, 40, 41**  
   \[
   9^2 + 40^2 = 81 + 1600 = 1681 = 41^2
   \]
   So, \((9, 40, 41)\) is yet another set of side lengths.

### Conclusion

To confirm that Jerry's triangle is a right triangle, the side lengths he provided should satisfy the Pythagorean theorem. For example, if Jerry's triangle has sides of 3, 4, and 5, or 5, 12, and 13, it would be a right triangle.

Would you like further details, or do you have any questions?  

Here are five related questions to expand on this topic:

1. What is the Pythagorean theorem, and how is it derived?
2. How can you use the Pythagorean theorem to find the missing side of a right triangle?
3. Are there infinitely many sets of integer sides that form right triangles?
4. What is the significance of Pythagorean triples in mathematics?
5. How can you determine if a triangle is a right triangle using its angles?

**Tip:** Always remember that for a triangle to be a right triangle, the side lengths must satisfy the Pythagorean theorem.

Learn how to verify if a triangle is a right triangle by checking if its side lengths satisfy the Pythagorean theorem. This guide provides examples of side lengths like (3, 4, 5) and (5, 12, 13) that form right triangles. Suitable for students in Grades 6-8.

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