Expert Verified Answer
Explanation:
The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. Symbolically, it can be written as:
a^2 + b^2 = c^2
where a and b are the lengths of the two shorter sides and c is the length of the hypotenuse.
To prove the Pythagorean theorem, you can use one of the many proofs that have been developed over the centuries. Here is one of the simplest proofs, known as the "triangle area proof":
Draw a right triangle with sides a, b, and c, where c is the length of the hypotenuse.
Draw squares on each of the three sides of the triangle, with sides equal to the lengths of the sides of the triangle.
The area of the square on the hypotenuse is c^2.
The area of each of the squares on the other two sides is a^2 and b^2.
The sum of the areas of the two smaller squares is a^2 + b^2.
The sum of the areas of the two smaller squares is equal to the area of the square on the hypotenuse, so a^2 + b^2 = c^2.
This proves the Pythagorean theorem.