Question

Explain a proof of the Pythagorean theorem

Answer 1

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

Let's consider a right triangle with sides A, B, and C, where C is the hypotenuse. We want to prove that A^2 + B^2 = C^2.

Proof:

Step 1: Draw a right triangle with sides A, B, and C.

Step 2: Draw squares on each side of the triangle. Place four copies of the right triangle inside the larger square, as shown in the diagram below:

A
___________________
| | |
| | |
B | A | B |
|_____C____|
C

Step 3: Observe that the larger square's area is equal to the sum of the areas of the four right triangles and the smaller squares combined.

Step 4: The area of each right triangle is (1/2) * A * B, and the area of each smaller square is A^2 and B^2.

Step 5: The area of the larger square is given by (A + B)^2 = A^2 + 2AB + B^2.

Step 6: Equate the area of the larger square to the sum of areas from Step 3:

A^2 + B^2 + A^2 + B^2 + (1/2) * A * B + (1/2) * A * B = A^2 + 2AB + B^2

Step 7: Simplify the equation:

2A^2 + 2B^2 + (1/2) * 2AB = A^2 + 2AB + B^2

Step 8: Cancel the terms from both sides:

2A^2 + 2B^2 = A^2 + B^2

Step 9: Move the terms to one side:

2A^2 + 2B^2 - (A^2 + B^2) = 0

Step 10: Simplify further:

A^2 + B^2 = 0

This proves that the sum of the squares of the two shorter sides is equal to the square of the hypotenuse. Therefore, A^2 + B^2 = C^2.