Algebra/The Pythagorean Theorem

If we name the two legs of a right triangle a and b, and the hypotenuse as c, then a² + b² = c². The following animation proves visually this fact.

For lots of other proofs (72 to be exact) see this page.

Pythagorean Triplets

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 Triplets. Common triplets 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 triplet, then all subsequent multiples of this triplet will satisfy the Pythagorean Theorem. This results in infinite more triplets:

6-8-10

9-12-15

12-16-20

10-24-26

And so on...

The Distance Formula

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 c. 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:

Template:TrigBoxOpen $(x_2-x_1{)}^2+(y_2-y_1{)}^2=x^2$ Template:TrigBoxClose

This formula is called the distance formula.

Another Forlmula is (and more simplified):

Template:TrigBoxOpen $a^2+b^2=c^2$ Template:TrigBoxClose Link title