There are several ways to prove the Pythagorean theorem. Here are two common methods:
1. Geometric Proof:
- Draw a right triangle ABC with side lengths a, b, and c (where c is the hypotenuse).
- Construct a square with side length (a + b) outside the triangle.
- Divide the square into four equal triangles by drawing lines connecting the midpoints of the square's sides.
- The area of the square is (a + b)^2, and the areas of the four triangles are (1/2)ab each.
- Thus, the area of the square is equal to the sum of the four triangle areas:
(a + b)^2 = 4 * (1/2)ab
(a + b)^2 = 2ab
a^2 + 2ab + b^2 = 2ab
a^2 + b^2 = c^2
2. Algebraic Proof:
- Start with the equation of a right triangle: a^2 + b^2 = c^2.
- Assume a, b, and c are positive integers.
- Square both sides of the equation: (a^2 + b^2)^2 = c^4.
- Expand the equation: a^4 + 2a^2b^2 + b^4 = c^4.
- Rewrite the equation in terms of squares: (a^2)^2 + 2(a^2)(b^2) + (b^2)^2 = c^4.
- Recognize that the equation represents the sum of two perfect squares: (a^2 + b^2)^2 = c^4.
- Take the square root of both sides: a^2 + b^2 = c^2.
These are just two examples of Pythagorean theorem proofs. There are many other proofs available, each with its own approach and variation.
Pythagorean theorem proofs quick check
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