In this image, you have a set of right triangle problems where you are asked to find the value of $x$. Some of the triangles require solving using the Pythagorean theorem:

$a^2 + b^2 = c^2$

where $c$ is the hypotenuse (the side opposite the right angle), and $a$ and $b$ are the legs of the triangle. Let’s go through some examples based on the layout:

Example 1: Find the value of $x$

1. For Triangle 1: Given sides 9 and 12, use the Pythagorean theorem to find $x$. $x^2 = 9^2 + 12^2 = 81 + 144 = 225 \implies x = \sqrt{225} = 15$ So, $x = 15$.

2. For Triangle 2: Given sides 9 and 15, use the Pythagorean theorem: $x^2 = 9^2 + 15^2 = 81 + 225 = 306 \implies x = \sqrt{306}$ $x \approx 17.5$.

3. For Triangle 3: Given the hypotenuse 65 and one leg 25, find $x$: $65^2 = 25^2 + x^2 \implies 4225 = 625 + x^2 \implies x^2 = 3600 \implies x = \sqrt{3600} = 60$ So, $x = 60$.

4. For Triangle 4: Given sides 3 and 4, use: $x^2 = 3^2 + 4^2 = 9 + 16 = 25 \implies x = \sqrt{25} = 5$ So, $x = 5$.

5. For Triangle 5: Given one leg 16 and hypotenuse 33, solve: $33^2 = 16^2 + x^2 \implies 1089 = 256 + x^2 \implies x^2 = 833 \implies x = \sqrt{833}$ $x \approx 28.9$.

6. For Triangle 6: Using the legs 11 and 28: $x^2 = 11^2 + 28^2 = 121 + 784 = 905 \implies x = \sqrt{905}$ $x \approx 30.1$.

Example 2: Use a Pythagorean triple

These problems involve known Pythagorean triples (e.g., 3-4-5, 5-12-13).

7. Triangle 7: A triple of $8, 15, 17$, so $x = 17$.

8. Triangle 8: Using the triple $24, 45, 51$, find $x = 51$.

9. Triangle 9: Using the triple $28, 96, x$, solve: $x = \sqrt{28^2 + 96^2} = \sqrt{784 + 9216} = \sqrt{10000} = 100$

10. Triangle 10: Using the triple $5, 12, 13$, so $x = 13$.

11. Triangle 11: Another common triple $8, 10, 6$, so $x = 6$.

12. Triangle 12: Using $12, 16, 20$, you have $x = 16$.

Would you like more details or explanations for any of these? Here are five related questions:

1. How can you identify Pythagorean triples?
2. What are some common Pythagorean triples to memorize?
3. Why is the hypotenuse always the longest side in a right triangle?
4. Can the Pythagorean theorem be used for non-right triangles?
5. What is the relationship between similar triangles and the Pythagorean theorem?

Tip: Memorizing common Pythagorean triples like (3, 4, 5), (5, 12, 13), and (7, 24, 25) can make solving these problems much faster.