Algebra/The Pythagorean Theorem

17.1: The Distance Formula

An Overview of The Pythagorean Theorem


The first expressions for mathematics were closer to Geometry than to Algebra, and the Greeks were the first people to study mathematics for its own sake, leading to generalized theories and proofs. A Greek mathematician named Euclid wrote a book called "The Elements" that presented Geometry as a logical system derived from 5 Axioms or givens and 5 "common sense" assertions about logic.

In the Euclid's Elements Proposition 47 is the Pythagorean Theorem which states "In any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle)." This is quite a mouthful, but as we will see below in algebra we can write it as $$a^2+b^2=c^2$$. The animation to the right shows an animation of Euclid's proof for Proposition 47.

The article on the Pythagorean Theorem shows several other approaches to prove the Pythagorean Theorem including one that was published by the future President of the United States James Garfield in 1876!

Integers that conform to the theorem are called Pythagorean Triples, which we will soon address. A clay tablet shows that Pythagorean triplets were being documented almost 4000 years ago, 1,000 years before Pythagoras. One archaeologist theorizes that the tablet might have been a problem set a teacher assigned a student.

The Distance Formula - Using The Pythagorean Theorem In Algebra
Let's say that there are two dots on a coordinate plane. How would you find the distance between the two without a ruler? Hint: draw a right triangle. Let's see if you can figure this out yourself before peeking!

Suppose you have two points, (x1, y1) and (x2, y2), and suppose that the length of the straight line between them is d. You can derive the distance formula by noticing that you can follow the following path between any two points to obtain a right triangle: start at point 1, change x (keep y constant) until you're directly above or below point 2, and then alter y and keep x constant until you're at point 2.

If you follow this path, the length of the first segment that you draw is $$ |x_2 - x_1| $$ and the length of the second is $$ |y_2 - y_1| $$. Also, since these two line segments form a right triangle, the Pythagorean Theorem applies and we can write $$(x_2 - x_1)^2 + (y_2 - y_1)^2 = d^2\, $$ or, solving for d,

$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

This formula is called the distance formula.

Another Formula is (and more simplified):

$$a^2+b^2=c^2$$

Pythagorean Triples
If there exist three positive integers who are sides of a right triangle (the sum of the squares of the 2 smaller integers is equal to the square of the largest), then the three numbers are called Pythagorean Triples. Common triples include:

3-4-5

5-12-13

7-24-25

8-15-17

12-35-37

20-21-29

Note: If the three numbers a-b-c are a Pythagorean Triple, then all subsequent multiples of this Triple will satisfy the Pythagorean Theorem.

Not all numbers create a Pythagorean Triple. In fact if you connect the corners of a square on a Cartesian grid you've just drawn a line that has an irrational length. $$ 1^2 + 1^2 = 2 $$ so the length of the line is $$ \sqrt 2 $$. It is said that the Greek mathematician who proved this, Hippasus was thrown off a ship and drowned for proving that the $$ \sqrt 2 $$ was irrational.

The Wikipedia article on Pythagorean Triples shows mathematical research on Pythagorean Triples. If you follow the links on the article you will find that modern mathematicians are still finding new patterns with them. Even if you don't find Pythagorean triples interesting for their own sake, it is worth memorizing the first set: 3-4-5. You are likely to encounter these numbers on standardized tests or in word problems. After you leave school you will find opportunities to use these lengths to quickly test whether an angle you are looking at is square.