State and prove the Pythagorean Theorem.
The Pythagorean Theorem states that in a right-angled 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.
To prove the Pythagorean Theorem, we need to start with the definitions of the terms involved. Let's consider a right-angled triangle with sides of lengths a, b, and c, where c is the hypotenuse.
Using the definition of a right-angled triangle, we know that the sum of the squares of the two shorter sides should be equal to the square of the hypotenuse:
a^2 + b^2 = c^2
To prove this, let's construct a square with side length c and four congruent right-angled triangles inside it, as shown below:
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The area of the large square can be calculated as c^2. Inside the large square, we have four congruent right-angled triangles, each with a base of a and height of b. The area of each triangle is (1/2) * a * b. Therefore, the total area of the four triangles is 4 * (1/2) * a * b = 2ab.
Since the area of the square is equal to the sum of the areas of the four triangles, we have:
c^2 = 2ab
Dividing both sides of the equation by 2, we get:
c^2/2 = ab
Now, let's express sides a and b as the sum of their squares:
a^2 = (a^2/2) + (a^2/2)
b^2 = (b^2/2) + (b^2/2)
Substituting these expressions into the equation c^2/2 = ab, we get:
(a^2/2) + (a^2/2) + (b^2/2) + (b^2/2) = c^2/2
Simplifying the equation, we have:
a^2 + b^2 = c^2
This equation is the Pythagorean Theorem. Therefore, we have proved that in a right-angled 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.