# Difference between revisions of "Pythagorean Theorem"

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− | The '''Pythagorean Theorem''' states that for a [[right triangle]] with legs of length <math>a</math> and <math>b</math> and [[hypotenuse]] of length <math>c</math> we have the relationship <math>{a}^{2}+{b}^{2}={c}^{2}</math>. This theorem has been know since antiquity and is a classic to prove; hundreds of proofs have been published and many can be demonstrated entirely visually. | + | The '''Pythagorean Theorem''' states that for a [[right triangle]] with legs of length <math>a</math> and <math>b</math> and [[hypotenuse]] of length <math>c</math> we have the relationship <math>{a}^{2}+{b}^{2}={c}^{2}</math>. This theorem has been know since antiquity and is a classic to prove; hundreds of proofs have been published and many can be demonstrated entirely visually(the book ''The Pythagorean Proposition'' alone consists of more than 370). The Pythagorean Theorem is one of the most frequently used theorems in [[geometry]], and is one of the many tools in a good geometer's arsenal. A very large number of geometry problems can be solved by building right triangles and applying the Pythagorean Theorem. |

This is generalized by the [[Geometric inequality#Pythagorean_Inequality | Pythagorean Inequality]] and the [[Law of Cosines]]. | This is generalized by the [[Geometric inequality#Pythagorean_Inequality | Pythagorean Inequality]] and the [[Law of Cosines]]. | ||

Line 13: | Line 13: | ||

Let <math>H </math> be the perpendicular to side <math>AB </math> from <math>{} C </math>. | Let <math>H </math> be the perpendicular to side <math>AB </math> from <math>{} C </math>. | ||

− | <center> | + | <center> |

+ | <asy> | ||

+ | pair A, B, C, H; | ||

+ | A = (0, 0); | ||

+ | B = (4, 3); | ||

+ | C = (4, 0); | ||

+ | H = foot(C, A, B); | ||

+ | |||

+ | draw(A--B--C--cycle); | ||

+ | draw(C--H); | ||

+ | draw(rightanglemark(A, C, B)); | ||

+ | draw(rightanglemark(C, H, B)); | ||

+ | label("$A$", A, SSW); | ||

+ | label("$B$", B, ENE); | ||

+ | label("$C$", C, SE); | ||

+ | label("$H$", H, NNW); | ||

+ | </asy> | ||

+ | </center> | ||

− | Since <math>ABC, CBH, ACH </math> are similar right triangles, and the areas of similar triangles are | + | Since <math>ABC, CBH, ACH</math> are similar right triangles, and the areas of similar triangles are proportional to the squares of corresponding side lengths, |

<center> | <center> | ||

<math> \frac{[ABC]}{AB^2} = \frac{[CBH]}{CB^2} = \frac{[ACH]}{AC^2} </math>. | <math> \frac{[ABC]}{AB^2} = \frac{[CBH]}{CB^2} = \frac{[ACH]}{AC^2} </math>. | ||

Line 27: | Line 44: | ||

<center>[[Image:Pyth2.png]]</center> | <center>[[Image:Pyth2.png]]</center> | ||

− | Evidently, <math>AY = AB - BC </math> and <math>AX = AB + BC </math>. By considering the [[ | + | Evidently, <math>AY = AB - BC </math> and <math>AX = AB + BC </math>. By considering the [[Power of a Point | power of point]] <math>A </math> with respect to <math>\omega </math>, we see |

<center> | <center> | ||

<math>AC^2 = AY \cdot AX = (AB-BC)(AB+BC) = AB^2 - BC^2 </math>. {{Halmos}} | <math>AC^2 = AY \cdot AX = (AB-BC)(AB+BC) = AB^2 - BC^2 </math>. {{Halmos}} | ||

</center> | </center> | ||

+ | |||

+ | === Proof 3 === | ||

+ | |||

+ | <math>ABCD</math> and <math>EFGH</math> are squares. | ||

+ | <center> | ||

+ | <asy> | ||

+ | pair A, B,C,D; | ||

+ | A = (-10,10); | ||

+ | B = (10,10); | ||

+ | C = (10,-10); | ||

+ | D = (-10,-10); | ||

+ | |||

+ | pair E,F,G,H; | ||

+ | E = (7,10); | ||

+ | F = (10, -7); | ||

+ | G = (-7, -10); | ||

+ | H = (-10, 7); | ||

+ | |||

+ | draw(A--B--C--D--cycle); | ||

+ | label("$A$", A, NNW); | ||

+ | label("$B$", B, ENE); | ||

+ | label("$C$", C, ESE); | ||

+ | label("$D$", D, SSW); | ||

+ | |||

+ | draw(E--F--G--H--cycle); | ||

+ | label("$E$", E, N); | ||

+ | label("$F$", F,SE); | ||

+ | label("$G$", G, S); | ||

+ | label("$H$", H, W); | ||

+ | |||

+ | label("a", A--B,N); | ||

+ | label("a", B--F,SE); | ||

+ | label("a", C--G,S); | ||

+ | label("a", H--D,W); | ||

+ | label("b", E--B,N); | ||

+ | label("b", F--C,SE); | ||

+ | label("b", G--D,S); | ||

+ | label("b", A--H,W); | ||

+ | label("c", E--H,NW); | ||

+ | label("c", E--F); | ||

+ | label("c", F--G,SE); | ||

+ | label("c", G--H,SW); | ||

+ | </asy> | ||

+ | </center> | ||

+ | <math>(a+b)^2=c^2+4\left(\frac{1}{2}ab\right)\implies a^2+2ab+b^2=c^2+2ab\implies a^2 + b^2=c^2</math>. {{Halmos}} | ||

== Common Pythagorean Triples == | == Common Pythagorean Triples == | ||

Line 41: | Line 103: | ||

<cmath>8-15-17</cmath> | <cmath>8-15-17</cmath> | ||

<cmath>9-40-41</cmath> | <cmath>9-40-41</cmath> | ||

+ | <cmath>12-35-37</cmath> | ||

<cmath>20-21-29</cmath> | <cmath>20-21-29</cmath> | ||

+ | <cmath>11-60-61</cmath> | ||

+ | |||

+ | |||

+ | Also Pythagorean Triples can be created with the a Pythagorean triple by multiplying the lengths by any integer. | ||

+ | For example, | ||

+ | <cmath>6-8-10 = (3-4-5)*2</cmath> | ||

+ | <cmath>21-72-75 = (7-24-25)*3</cmath> | ||

+ | <cmath>10-24-26 = (5-12-13)*2</cmath> | ||

== Problems == | == Problems == | ||

Line 47: | Line 118: | ||

* [[2006_AIME_I_Problems/Problem_1 | 2006 AIME I Problem 1]] | * [[2006_AIME_I_Problems/Problem_1 | 2006 AIME I Problem 1]] | ||

* [[2007 AMC 12A Problems/Problem 10 | 2007 AMC 12A Problem 10]] | * [[2007 AMC 12A Problems/Problem 10 | 2007 AMC 12A Problem 10]] | ||

+ | === Sample Problem === | ||

+ | Right triangle <math>ABC</math> has legs of length <math>333</math> and <math>444</math>. Find the hypotenuse of <math>ABC</math>. | ||

+ | ==== Solution 1 (Bash) ==== | ||

+ | <math>\sqrt{333^2 + 444^2} = 555</math>. | ||

+ | ==== Solution 2 (Using 3-4-5) ==== | ||

+ | We see <math>333-444</math> looks like the legs of a <math>3-4-5</math> right triangle with a multiplication factor of 111. Thus <math>5*111 = 555</math>. | ||

+ | |||

+ | === Another Problem === | ||

+ | Right triangle <math>ABC</math> has side lengths of <math>3</math> and <math>4</math>. Find the sum of all the possible hypotenuses. | ||

+ | ==== Solution (Casework) ==== | ||

+ | Case 1: 3 and 4 are the legs. Then 5 is the hypotenuse. | ||

+ | Case 2: 3 is a leg and 4 is the hypotenuse. | ||

+ | There are no more cases as the hypotenuse has to be greater than the leg. | ||

+ | This makes the sum <math>4+5=9</math>. | ||

== External links == | == External links == | ||

− | *[http://www.cut-the-knot.org/pythagoras/index.shtml | + | *[http://www.cut-the-knot.org/pythagoras/index.shtml 118 proofs of the Pythagorean Theorem] |

[[Category:Geometry]] | [[Category:Geometry]] | ||

[[Category:Theorems]] | [[Category:Theorems]] | ||

+ | |||

+ | [[Category:Geometry]] |

## Latest revision as of 18:38, 26 January 2020

The **Pythagorean Theorem** states that for a right triangle with legs of length and and hypotenuse of length we have the relationship . This theorem has been know since antiquity and is a classic to prove; hundreds of proofs have been published and many can be demonstrated entirely visually(the book *The Pythagorean Proposition* alone consists of more than 370). The Pythagorean Theorem is one of the most frequently used theorems in geometry, and is one of the many tools in a good geometer's arsenal. A very large number of geometry problems can be solved by building right triangles and applying the Pythagorean Theorem.

This is generalized by the Pythagorean Inequality and the Law of Cosines.

## Contents

## Proofs

In these proofs, we will let be any right triangle with a right angle at .

### Proof 1

We use to denote the area of triangle .

Let be the perpendicular to side from .

Since are similar right triangles, and the areas of similar triangles are proportional to the squares of corresponding side lengths,

.

But since triangle is composed of triangles and , , so . ∎

### Proof 2

Consider a circle with center and radius . Since and are perpendicular, is tangent to . Let the line meet at and , as shown in the diagram:

Evidently, and . By considering the power of point with respect to , we see

. ∎

### Proof 3

and are squares.

. ∎

## Common Pythagorean Triples

A Pythagorean Triple is a set of 3 positive integers such that , i.e. the 3 numbers can be the lengths of the sides of a right triangle. Among these, the Primitive Pythagorean Triples, those in which the three numbers have no common divisor, are most interesting. A few of them are:

Also Pythagorean Triples can be created with the a Pythagorean triple by multiplying the lengths by any integer.
For example,

## Problems

### Introductory

### Sample Problem

Right triangle has legs of length and . Find the hypotenuse of .

#### Solution 1 (Bash)

.

#### Solution 2 (Using 3-4-5)

We see looks like the legs of a right triangle with a multiplication factor of 111. Thus .

### Another Problem

Right triangle has side lengths of and . Find the sum of all the possible hypotenuses.

#### Solution (Casework)

Case 1: 3 and 4 are the legs. Then 5 is the hypotenuse. Case 2: 3 is a leg and 4 is the hypotenuse. There are no more cases as the hypotenuse has to be greater than the leg. This makes the sum .