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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.

Find the value of x in each right triangle using the Pythagorean theorem and Pythagorean triples.

Learn how to find the value of x in right triangles by applying the Pythagorean theorem and identifying Pythagorean triples. This lesson includes step-by-step solutions for various triangle configurations, including both basic and complex problems. Suitable for Grades 8-10 students.

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