1. **State the problem:** We need to determine if a triangle with sides $a$, $b$, and $c$ is a right triangle using the Pythagorean theorem. 2. **Recall the Pythagorean theorem:** For a right triangle, the sum of the squares of the two shorter sides equals the square of the longest side: $$a^2 + b^2 = c^2$$ 3. **Given:** $$a^2 + b^2 = 410$$ 4. **Find $c^2$:** To check if the triangle is right-angled, we need the value of $c^2$ (the square of the longest side). Since $c$ is the longest side, if $c^2 = 410$, then the triangle is right-angled. 5. **Conclusion:** Without the value of $c^2$, we cannot definitively say if the triangle is right-angled. If you provide $c^2$, compare it to 410: - If $c^2 = 410$, the triangle is right-angled. - If $c^2 \neq 410$, it is not a right triangle.