The triangle in your image appears to be a right triangle. Here's what I see:

• The vertical leg $PT = 9$.
• The horizontal leg $PS = 12$.
• The hypotenuse $TS$ is labeled as $x$.

To solve for $x$, we can use the Pythagorean theorem: $x^2 = 9^2 + 12^2$

Step-by-step solution:

1. Compute the squares of the legs: $9^2 = 81 \quad \text{and} \quad 12^2 = 144$

2. Add them together: $x^2 = 81 + 144 = 225$

3. Take the square root to find $x$: $x = \sqrt{225} = 15$

Final Answer:

$x = 15$

Would you like a detailed explanation of the theorem or additional examples?

Here are 5 related questions to extend your understanding:

1. What are some real-world applications of the Pythagorean theorem?
2. How do you determine whether a triangle is a right triangle using side lengths?
3. Can the Pythagorean theorem be used in three dimensions?
4. What happens if one of the legs is missing? How would you solve for it?
5. How is the Pythagorean theorem related to trigonometric functions?

Tip: Always ensure the triangle is a right triangle before applying the Pythagorean theorem!