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 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. | 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 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. | ||
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== Proofs == | == Proofs == | ||
− | In these proofs, we will let <math> | + | In these proofs, we will let <math>ABC </math> be any right triangle with a right angle at <math>{} C </math>. |
=== Proof 1 === | === Proof 1 === | ||
− | We use <math> | + | We use <math>[ABC] </math> to denote the area of triangle <math>ABC </math>. |
− | Let <math> | + | Let <math>H </math> be the perpendicular to side <math>AB </math> from <math>{} C </math>. |
<center>[[Image:Pyth1.png]]</center> | <center>[[Image:Pyth1.png]]</center> | ||
− | Since <math> | + | Since <math>ABC, CBF, ACF </math> are similar right triangles, and the areas of similar triangles are proportionate to the squares of corresponding side lengths, |
<center> | <center> | ||
<math> \frac{[ABC]}{AB^2} = \frac{[CBF]}{CB^2} = \frac{[ACF]}{AC^2} </math>. | <math> \frac{[ABC]}{AB^2} = \frac{[CBF]}{CB^2} = \frac{[ACF]}{AC^2} </math>. | ||
</center> | </center> | ||
− | But since triangle <math> | + | But since triangle <math>ABC </math> is composed of triangles <math>CBF </math> and <math>ACF </math>, <math>[ABC] = [CBF] + [ACF] </math>, so <math>AB^2 = CB^2 + AC^2 </math>. {{Halmos}} |
=== Proof 2 === | === Proof 2 === | ||
− | Consider a circle <math> | + | Consider a circle <math>\omega </math> with center <math>B </math> and radius <math>BC </math>. Since <math>BC </math> and <math>AC </math> are perpendicular, <math>AC </math> is tangent to <math>\omega </math>. Let the line <math>AB </math> meet <math>\omega </math> at <math>Y </math> and <math>X </math>, as shown in the diagram: |
<center>[[Image:Pyth2.png]]</center> | <center>[[Image:Pyth2.png]]</center> | ||
− | Evidently, <math> | + | 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> | + | <math>AC^2 = AY \cdot AX = (AB-BC)(AB+BC) = AB^2 - BC^2 </math>. {{Halmos}} |
</center> | </center> | ||
Revision as of 21:35, 22 September 2007
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 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
[hide]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 proportionate 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
. ∎
Introductory
Example Problems
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:
3-4-5
5-12-13
7-24-25
8-15-17
9-40-41