determine if a 3-4-5 triangle is a right triangle

1. Understand the Pythagorean Theorem: A triangle is a right triangle if and only if the sum of the squares of the two shorter sides equals the square of the longest side. This is expressed as $a^2 + b^2 = c^2$, where $c$ is the longest side (hypotenuse).

1. Identify the sides: The given sides of the triangle are $3$, $4$, and $5$. Here, $a = 3$, $b = 4$, and $c = 5$.

3. Apply the theorem: Calculate the squares of the sides:
$$a^2 = 3^2 = 9$$
$$b^2 = 4^2 = 16$$
$$c^2 = 5^2 = 25$$

4. Check the equality: Add the squares of the two shorter sides and compare to the square of the longest side:
$$a^2 + b^2 = 9 + 16 = 25$$
Since $25 = 25$, the condition $a^2 + b^2 = c^2$ is satisfied.

1. Conclusion: Because the side lengths satisfy the Pythagorean theorem, the triangle must be a right triangle.