Final answer:
The Pythagorean Theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. It can be proven with numbers and a diagram by considering a specific right triangle and showing that the equation holds true.
Step-by-step explanation:
The Pythagorean Theorem is a fundamental concept in geometry that relates the sides of a right-angled triangle. It states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This can be represented by the equation a² + b² = c², where a and b are the lengths of the legs of the triangle and c is the length of the hypotenuse.
Let's prove the Pythagorean Theorem with numbers and a diagram:
Consider a right triangle with side lengths a = 3 and b = 4 units. We can calculate the squares of these side lengths: a² = 3² = 9 and b² = 4² = 16. According to the Pythagorean Theorem, the square of the hypotenuse c² is equal to the sum of the squares of a and b, which is 9 + 16 = 25. Taking the square root of both sides, we find that c = √25 = 5. Therefore, the length of the hypotenuse is 5 units, confirming the Pythagorean Theorem.