Question

Area of the inner square = c² area of the 4 triangles = a divide by the area of the big square = make and solve an equation using the 3 areas you found to prove the Pythagorean theorem.

Answer 1

The Pythagorean Theorem, a² + b² = c², is proven by comparing the areas of a large square, an inner square, and four right triangles inside the large square, confirming that the sum of the areas of the squares on the triangle's legs equals the area of the square on the hypotenuse.

Step-by-step explanation:

The Pythagorean Theorem states that in a right-angled triangle, the area of the square on the hypotenuse (c) is equal to the sum of the areas of the squares on the other two sides (a and b). This can be written as a² + b² = c². To prove this, consider a large square with a side length of a + b, which contains four identical right triangles with legs a and b, and an inner square with sides of length c. The area of the large square is (a + b)² and the area of the inner square is c².

The area of the large square is also equal to the area of the inner square plus the area of the four triangles. Since the area of one triangle is ½ab, the area of the four triangles is 4 × ½ab or 2ab. Our equation using the areas is therefore (a + b)² = c² + 2ab, which simplifies to a² + 2ab + b² = c² + 2ab. Subtracting 2ab from both sides, we are left with a² + b² = c², thus proving the Pythagorean Theorem.