The question asks to determine which group of side lengths would form a right triangle based on the given diagram. Using the Pythagorean Theorem, $a^2 + b^2 = c^2$, we can test each group of side lengths.

Let's evaluate each option:

1. $20, 21, 29$

• $20^2 + 21^2 = 29^2$?
• $400 + 441 = 841$
• $841 = 841$: Correct

2. $7, 24, 27$

• $7^2 + 24^2 = 27^2$?
• $49 + 576 = 625$
• $625 \neq 729$: Incorrect

3. $11, 32, 34$

• $11^2 + 32^2 = 34^2$?
• $121 + 1024 = 1156$
• $1145 \neq 1156$: Incorrect

4. $12, 35, 37$

• $12^2 + 35^2 = 37^2$?
• $144 + 1225 = 1369$
• $1369 = 1369$: Correct

5. $8, 15, 19$

• $8^2 + 15^2 = 19^2$?
• $64 + 225 = 289$
• $289 \neq 361$: Incorrect

Correct Answers:

• $20, 21, 29$
• $12, 35, 37$

Would you like me to go into further detail or clarify anything?

Here are five additional questions you might find interesting:

1. How does the Pythagorean Theorem apply to non-right triangles?
2. Can irrational numbers form a Pythagorean triple?
3. What are primitive Pythagorean triples, and how are they generated?
4. How does the Pythagorean Theorem extend to three dimensions (e.g., a rectangular prism)?
5. How is the Pythagorean Theorem used in real-world applications?

Tip: When working with the Pythagorean Theorem, always verify calculations carefully, as small errors can make a big difference!