Consider a right-angled triangle with sides of lengths a, b, and c, where c is the hypotenuse (the side opposite the right angle).
Step 1: Draw a square with side length (a + b). This square can be divided into four congruent right-angled triangles and a smaller square at the center.
Step 2: The area of the larger square is (a + b)^2 since it has side length (a + b).
Step 3: Each of the four right-angled triangles has an area of (1/2)ab since each triangle can be formed by arranging two congruent right-angled triangles with sides a and b.
Step 4: The area of the smaller square at the center is c^2 since it has side length c (the hypotenuse of the right-angled triangle).
Step 5: Since the larger square is made up of the four triangles and the smaller square, we can express its area as the sum of these parts:
(a + b)^2 = 4 * (1/2)ab + c^2
Step 6: Simplify the equation:
(a^2 + 2ab + b^2) = 2ab + c^2
Step 7: Eliminate the 2ab term on both sides:
a^2 + b^2 = c^2
Step 8: This equation shows that the sum of the squares of the two legs (a^2 + b^2) is equal to the square of the hypotenuse (c^2), which is the Pythagorean Theorem.
Therefore, the Pythagorean Theorem is proven. The squares of the lengths of the two legs of a right-angled triangle always add up to the square of the length of the hypotenuse.
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