To find the length of the third side of this right triangle, we can use 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 two other sides. Here, we know:

• One leg $a = 13$
• The hypotenuse $c = 15$

We need to find the other leg $b$.

Step-by-Step Solution

1. Substitute the values we know into the Pythagorean theorem: $13^2 + b^2 = 15^2$

2. Calculate $13^2$ and $15^2$: $169 + b^2 = 225$

3. Subtract 169 from both sides to isolate $b^2$: $b^2 = 225 - 169$ $b^2 = 56$

4. Take the square root of both sides to solve for $b$: $b = \sqrt{56} \approx 7.5$

Answer

The length of the third side is approximately 7.5, rounded to the nearest tenth.

Would you like further explanation, or do you have any questions?

Here are five related questions to expand your understanding:

1. How would the Pythagorean theorem be applied if all three sides were unknown but one angle was 90 degrees?
2. What if only the hypotenuse and one angle (not 90 degrees) were known? How would we find the other sides?
3. Can the Pythagorean theorem be used in non-right triangles? If not, what formulas apply there?
4. If the hypotenuse were longer, say 20, what would the length of the third side be with one side remaining 13?
5. How can the Pythagorean theorem help in real-life applications, like in construction or navigation?

Tip: When working with right triangles, the Pythagorean theorem is an essential tool to find unknown side lengths, as long as you know two of the three sides.