The triangle in the image is a right triangle with one side labeled 5, the hypotenuse labeled 13, and we are tasked with finding the length of the third side.

To solve for the missing side, we can use the Pythagorean theorem:

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

Where:

• $a$ and $b$ are the legs of the triangle,
• $c$ is the hypotenuse (the side opposite the right angle).

Given:

• One leg $a = 5$,
• Hypotenuse $c = 13$,

We need to find the other leg $b$. Plugging the values into the Pythagorean theorem:

$5^2 + b^2 = 13^2$

$25 + b^2 = 169$

$b^2 = 169 - 25$

$b^2 = 144$

$b = \sqrt{144}$

$b = 12$

Thus, the length of the third side is 12.

Would you like further details or clarifications?

Related Questions:

1. How would the Pythagorean theorem apply if we were given the other two sides instead?
2. What if the triangle was not a right triangle? Which formula would we use then?
3. What are the conditions under which the Pythagorean theorem is valid?
4. How does the Pythagorean theorem help in determining distances in real-world problems?
5. Can we use trigonometric functions to find the sides of a right triangle as well?

Tip: In any right triangle, the hypotenuse is always the longest side, and the Pythagorean theorem only works with right triangles.