Problem Analysis Branch of Mathematics: Geometry Question Type: Proof Given Information A right-angled triangle has: Legs (perpendicular sides): and Hypotenuse: To Prove a^2+b^2=c^2 This is the Pythagorean Theorem. --- Proof (Using Similar Triangles) Consider a right triangle with . Let: Draw the altitude from the right angle to the hypotenuse . Let: Hence, x+y=c. Step 1: Use Similar Triangles The three triangles , , and are similar. From the similarity of and , \frac{b}{c}=\frac{x}{b}. Therefore, b^2=cx. Similarly, from the similarity of and , \frac{a}{c}=\frac{y}{a}. Hence, a^2=cy. Step 2: Add the Two Equations Adding the two results, a^2+b^2=cx+cy. Factor out : a^2+b^2=c(x+y). Since x+y=c, we obtain a^2+b^2=c(c)=c^2. Thus, \boxed{a^2+b^2=c^2.} --- Conclusion Therefore, in every right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. \boxed{a^2+b^2=c^2} Hence, the Pythagorean Theorem is proved. --- Final Answer \boxed{a^2+b^2=c^2} The Pythagorean Theorem has been proved.