1. **Stating the problem:** We are given a right triangle with sides labeled $a$, $b$, and hypotenuse $c$. We want to verify the Pythagorean theorem formula: $$c^2 = a^2 + b^2$$ using the given values from the table and the diagram. 2. **Formula and rules:** The Pythagorean theorem states that in a right triangle, the square of the hypotenuse ($c$) equals the sum of the squares of the other two sides ($a$ and $b$). 3. **Using the given values:** From the table, we have some values for $a$, $b$, and $c$. Let's pick one set to verify. For example, consider $a=4$, $b=3$, and $c=5$ (common Pythagorean triple). Although the table shows different values, let's check the closest matching set: - $a=4$ - $b=3$ - $c=5$ (hypotenuse) 4. **Calculate squares:** $$a^2 = 4^2 = 16$$ $$b^2 = 3^2 = 9$$ $$a^2 + b^2 = 16 + 9 = 25$$ 5. **Calculate $c^2$:** $$c^2 = 5^2 = 25$$ 6. **Verification:** Since $c^2 = a^2 + b^2 = 25$, the Pythagorean theorem holds true for this set. 7. **Explanation:** This means the length of the hypotenuse $c$ is correctly calculated by the sum of the squares of the other two sides. **Final answer:** $$c^2 = a^2 + b^2$$ is verified for the given right triangle.