# Geometry/Chapter 14

## The Pythagorean Theorem

editThe 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.

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

editNow that we know the Pythagorean Theorem, take a look at the following 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 a^{2}+2ab++b^{2}=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:

*Summary*: The Pythagorean Theorem can be proved using diagrams.

## Exercise

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- Geometry Main Page
- Motivation
- Introduction
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- Geometry/Chapter 1/Lesson 1 Introduction
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- Geometry/Chapter 1/Lesson 3 Undefined Terms
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- Geometry/Chapter 1/Lesson 5 Theorems
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- Geometry/Chapter 2 Proofs
- Geometry/Chapter 3 Logical Arguments
- Geometry/Chapter 4 Congruence and Similarity
- Geometry/Chapter 5 Triangle: Congruence and Similiarity
- Geometry/Chapter 6 Triangle: Inequality Theorem
- Geometry/Chapter 7 Parallel Lines, Quadrilaterals, and Circles
- Geometry/Chapter 8 Perimeters, Areas, Volumes
- Geometry/Chapter 9 Prisms, Pyramids, Spheres
- Geometry/Chapter 10 Polygons
- Geometry/Chapter 11
- Geometry/Chapter 12 Angles: Interior and Exterior
- Geometry/Chapter 13 Angles: Complementary, Supplementary, Vertical
- Geometry/Chapter 14 Pythagorean Theorem: Proof
- Geometry/Chapter 15 Pythagorean Theorem: Distance and Triangles
- Geometry/Chapter 16 Constructions
- Geometry/Chapter 17 Coordinate Geometry
- Geometry/Chapter 18 Trigonometry
- Geometry/Chapter 19 Trigonometry: Solving Triangles
- Geometry/Chapter 20 Special Right Triangles
- Geometry/Chapter 21 Chords, Secants, Tangents, Inscribed Angles, Circumscribed Angles
- Geometry/Chapter 22 Rigid Motion
- Geometry/Appendix A Formulae
- Geometry/Appendix B Answers to problems
- Appendix C. Geometry/Postulates & Definitions
- Appendix D. Geometry/The SMSG Postulates for Euclidean Geometry