To find the length of the third side in this right triangle, we can use the Pythagorean Theorem, which states:

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

where $c$ is the hypotenuse (the longest side opposite the right angle), and $a$ and $b$ are the other two sides.

Here, the sides provided are:

• One leg $a = 13$
• Hypotenuse $c = 15$

We need to find the length of the missing leg, $b$.

Solution

1. Substitute the values into the Pythagorean Theorem:

$13^2 + b^2 = 15^2$

2. Calculate the squares:

$169 + b^2 = 225$

3. Solve for $b^2$:

$b^2 = 225 - 169$ $b^2 = 56$

4. Take the square root of both sides:

$b = \sqrt{56} \approx 7.5$

Answer

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

Would you like a more detailed explanation or have any questions?

Here are some related questions for further practice:

1. How would the solution change if the hypotenuse were unknown?
2. What if both legs were given as 9 and 12? What would the hypotenuse be?
3. How would we verify if a triangle is right-angled given three sides?
4. Can we use trigonometry to find an angle in this triangle with the sides given?
5. What if the triangle sides were in a different proportion—would it still be right-angled?

Tip: For right triangles, identifying the hypotenuse correctly is crucial since it’s always opposite the right angle and the longest side.