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Geometry/Chapter 14

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The Pythagorean Theorem

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The Pythagorean Theorem shows the relationship between the sides (a and b) and the hypotenuse (c) of a right triangle. The right triangle I will be using is shown below.

 
A Right Triangle

The Pythagorean Theorem states that, in a right triangle,the square of a (a²) plus the square of b (b²) is equal to the square of c (c²).

 


Summary: The Pythagorean Theorem is a²+b²=c², or leg² + leg² = hyp². It works only for right triangles.

Proof of the Pythagorean Theorem

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Now that we know the Pythagorean Theorem, take a look at the following diagram.

 
A diagram.

Look at the large square. The large square's area can be written as:

(a+b)(a+b)

or as:

(a+b)²

since each side's length is (a+b). Look at the tilted square in the middle. Its area can be written as:

c².

Now, look at each of the triangles at the corners of the large square. Each triangle's area is:

½ab

There are four triangles, so the area of all four of them combined is:

4(½ab)

The area of the large square is equal to the area of the four triangles plus the area of the tilted square. This can be written as:

(a+b)²=c²+4(½ab)

Using Algebra, this can be simplified. 

      (a+b)²=c²+4(½ab)
      (a+b)(a+b)=c²+2ab  
      a2+2ab++b2=c²+2ab

 -2ab      -2ab

a²+b²=c²

Now we can see why the Pythagorean Theorem works, or, in other words, we can see proof of the Pythagorean Theorem.

However, this proof is not based on Euclidean Geometry. It is not elementary.

There are thousands more proofs of the Pythagorean theorem, too.

  • You should be able to explain why the following is proof of the Pythagorean Theorem:
 
Another proof of the Pythagorean Theorem.

Summary: The Pythagorean Theorem can be proved using diagrams.

Exercise

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