1. **State the problem:** We want to prove the Pythagorean Theorem, which states that for a right triangle with legs $a$ and $b$ and hypotenuse $c$, the equation $$a^2 + b^2 = c^2$$ holds true. 2. **Given values:** $a=5$, $b=12$, and $c=13$. 3. **Check the equation for these values:** Calculate $a^2 + b^2$ and compare it to $c^2$. 4. Calculate $a^2 + b^2$: $$5^2 + 12^2 = 25 + 144 = 169$$ 5. Calculate $c^2$: $$13^2 = 169$$ 6. Since $a^2 + b^2 = c^2$ holds true for $a=5$, $b=12$, and $c=13$, this supports the Pythagorean Theorem. 7. **Check the equation for another set:** $a=7$, $b=24$, $c=25$. 8. Calculate $a^2 + b^2$: $$7^2 + 24^2 = 49 + 576 = 625$$ 9. Calculate $c^2$: $$25^2 = 625$$ 10. Since $a^2 + b^2 = c^2$ also holds true for $a=7$, $b=24$, and $c=25$, this further supports the theorem. 11. **Conclusion:** Showing that $a^2 + b^2 = c^2$ holds for these specific triples is a way to verify the theorem for those cases, but the full proof requires a general argument. 12. **Answer to the multiple-choice question:** The option "Set two different expressions for the area of a figure equal to each other and manipulate algebraically to show $a^2 + b^2 = c^2$" is the method that would prove the Pythagorean Theorem. This is because it uses a general algebraic proof rather than just verifying specific cases or counterexamples.