To solve for the side length $x$ in the given right triangle, we can apply the Pythagorean theorem:

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

Here:

• $a = 9$ (one leg),
• $c = 10$ (hypotenuse),
• $b = x$ (the other leg we are solving for).

Step-by-step solution:

1. Substitute the known values into the formula: $9^2 + x^2 = 10^2$

2. Calculate the squares: $81 + x^2 = 100$

3. Isolate $x^2$: $x^2 = 100 - 81$

4. Simplify: $x^2 = 19$

5. Take the square root of both sides: $x = \sqrt{19}$

6. Approximate the square root to two decimal places: $x \approx 4.36$

Final Answer:

$x \approx 4.36$

Let me know if you'd like further clarification!

Expanding Questions:

1. How would the Pythagorean theorem change if the hypotenuse was unknown?
2. Can you describe a real-life scenario where the Pythagorean theorem might be used?
3. What are the properties of right triangles that make this formula applicable?
4. How is this theorem extended in three dimensions (e.g., distance formula)?
5. Why does the square root operation yield two possible solutions, and why do we pick the positive root in this case?

Tip:

Always ensure that the hypotenuse (the longest side) is labeled correctly in right triangle problems when using the Pythagorean theorem.