what is a proof of the pythagoarean theorem?
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Imagine having 3 perfectly square boxes. Perfectly square meaning the Length(L) and Width(W) of each indivudal box are the same but of different total areas. Arrange them so that one of the boxes represents the "A" dimension of a right angle triangle. Another Box representing the "B" dimension of a right triangle and the last box representing the "C" dimension (hypotneuse) of the configured triangle. Looking at the individual boxes areas (L x W), a clear pattern exists. Since these boxes are perfectly square (L = W) we can say A^2 + B^2 = C^2. Meaning the area of "box C" is equal to the area of "box A" plus the area of "box B"
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 other two sides. This theorem has been proven using various methods throughout history.
One of the most common methods to prove the Pythagorean theorem is the geometric proof. Here's a step-by-step explanation of how to prove it:
Step 1: Start with a right-angled triangle with sides of lengths a, b, and c, where c is the length of the hypotenuse.
Step 2: Cut out four copies of the right-angled triangle. Arrange them as shown, forming a larger square with side length (a+b).
Step 3: Break down the large square into smaller squares and two congruent rectangles.
Step 4: In the figure, you can see that the area of the larger square is equal to the sum of the areas of the four smaller squares and the two rectangles.
Step 5: One of the smaller squares has a side length of a, and another smaller square has a side length of b. So their respective areas are a^2 and b^2.
Step 6: The two congruent rectangles have sides of length a and b, so their areas are a*b and a*b.
Step 7: The remaining square in the center has sides of length c, so its area is c^2.
Step 8: Summing up the areas, we have a^2 + b^2 + a*b + a*b = (a+b)^2.
Step 9: Simplifying the equation, we have a^2 + b^2 + 2ab = a^2 + 2ab + b^2.
Step 10: Canceling out the common terms on both sides of the equation, we are left with 2ab = 2ab.
Step 11: This shows that the area of the large square is equal to the sum of the areas of the smaller squares and rectangles.
Step 12: Thus, (a+b)^2 = a^2 + b^2 + c^2.
Step 13: Now, if you subtract the common terms (a^2 + b^2) from both sides of the equation, you will be left with c^2 = a^2 + b^2.
Step 14: And that is the Pythagorean theorem in its algebraic form, proving that the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Please note that this is just one of the geometric proofs of the Pythagorean theorem. There are other methods, including algebraic and trigonometric proofs, that provide different perspectives on the theorem.