How to Prove the Pythagorean Theorem
Draw four congruent right triangles., Arrange the triangles so that they form a square with sides a+b., Rearrange the same four triangles such that they form two equal rectangles inside a larger square., Recognize that the area not formed by the...
Step-by-Step Guide
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Step 1: Draw four congruent right triangles.
Congruent triangles are ones that have three identical sides.
Designate the legs of length a and b and hypotenuse of length c.
The Pythagorean Theorem states that the sum of squares of the two legs of a right triangle is equal to the square of the hypotenuse, so we need to prove a2 + b2 = c2.
Remember, the Pythagorean Theorem only applies to right triangles. -
Step 2: Arrange the triangles so that they form a square with sides a+b.
With the triangles placed in this way, they will form a smaller square (in green) inside the larger square with four equal sides of length c, the hypotenuse of each triangle.The larger square has sides of length a+b.
You can rotate (turn) the entire arrangement by 90 degrees and it will be exactly the same.
You can repeat this as many times as you like.
This is only possible because the four angles at the corners are equal. , Again, the larger square will have sides of length a+b, but in this configuration there are two rectangles (in grey) of equal size and two smaller squares within the larger square.
The larger of the smaller squares (in red) has sides of length a, while the smaller square (in blue) has sides of length b.The hypotenuse of the original triangles is now the diagonal of the two rectangles formed by the triangles. , In both cases, you have a large square with sides of a+b.
Given this, the areas of both of the large squares are equal.
Looking at both arrangements, you can see that the total area of the green square must equal the areas of the red and blue squares added together in the second arrangement.
In both arrangements we partially covered the surface with exactly the same amount, four grey triangles that didn't overlap.
This means that also the area left out by the triangles must be equal in both arrangements.
Therefore, the area of the blue and the red square taken together must be equal to the area of the green square. , The blue area is a2, the red area, b2 and the green area, c2.
The red and blue squares must be added together to equal the area of the green square; therefore, blue area + red area = green area: a2 + b2 = c2.This finishes the proof. -
Step 3: Rearrange the same four triangles such that they form two equal rectangles inside a larger square.
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Step 4: Recognize that the area not formed by the triangles is equal in both arrangements.
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Step 5: Set the areas of each arrangement equal to each other.
Detailed Guide
Congruent triangles are ones that have three identical sides.
Designate the legs of length a and b and hypotenuse of length c.
The Pythagorean Theorem states that the sum of squares of the two legs of a right triangle is equal to the square of the hypotenuse, so we need to prove a2 + b2 = c2.
Remember, the Pythagorean Theorem only applies to right triangles.
With the triangles placed in this way, they will form a smaller square (in green) inside the larger square with four equal sides of length c, the hypotenuse of each triangle.The larger square has sides of length a+b.
You can rotate (turn) the entire arrangement by 90 degrees and it will be exactly the same.
You can repeat this as many times as you like.
This is only possible because the four angles at the corners are equal. , Again, the larger square will have sides of length a+b, but in this configuration there are two rectangles (in grey) of equal size and two smaller squares within the larger square.
The larger of the smaller squares (in red) has sides of length a, while the smaller square (in blue) has sides of length b.The hypotenuse of the original triangles is now the diagonal of the two rectangles formed by the triangles. , In both cases, you have a large square with sides of a+b.
Given this, the areas of both of the large squares are equal.
Looking at both arrangements, you can see that the total area of the green square must equal the areas of the red and blue squares added together in the second arrangement.
In both arrangements we partially covered the surface with exactly the same amount, four grey triangles that didn't overlap.
This means that also the area left out by the triangles must be equal in both arrangements.
Therefore, the area of the blue and the red square taken together must be equal to the area of the green square. , The blue area is a2, the red area, b2 and the green area, c2.
The red and blue squares must be added together to equal the area of the green square; therefore, blue area + red area = green area: a2 + b2 = c2.This finishes the proof.
About the Author
Kenneth Hill
Specializes in breaking down complex organization topics into simple steps.
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