Algebra/Chapter 17/The Pythagorean Theorem

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Algebra Chapter 17: Conic Sections Section 1: The Distance Formula

Loci of Points

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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 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,

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

This formula is called the distance formula.

Another Formula is (and more simplified):

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