In this lesson you will learn more about the “distance formula”, where it comes from, and how it is applied to different shapes (other than triangles).
First, let’s consider what you already know (from Algebra 1). You should have the formula for calculating a line’s slope (given any two point on the line), memorized. y y m 2 1 x2 x1
y2 y1
x2 x1
Now consider, a segment is part of a line, with endpoints on the line. So, just like calculating the slope of a line, you could just as easily calculate the slope of the segment. However, something else you can do with this same information is calculate a distance from one endpoint to another. Lines have no endpoints, that’s part of the reason this is not discussed much when introducing slope.
y2 y1
x2 x1
Let d = the distance between point A and point B. Using the Pythagorean Theorem, we know that:
d 2 x2 x1 y2 y1
2
2
B d
And, solving for d, (taking the square root of both sides), we get:
y2 y1
d
x2 x1 y2 y1
2
2
A
x2 x1
NOTE: The whole point of this is to help you realize that the distance formula is not some new creation! It utilizes parts from the slope formula and the Pythagorean Theorem.
The distance formula (Pythagorean Theorem) for shapes other than triangles: What about circles? We measure circumference and area of circles based on the circle’s radius. But what is the radius of a circle? Isn’t it a line segment, from the center (a point) to some point on the circle (another point)? Example: What is the radius of the circle who’s center is at coordinate 1,1 and passes through the point
4,5 ?
4,5
r
y2 y1
1,1
x2 x1
Using the applet below, left-click and drag points A and B around to different locations and calculate the distance between them. Check your answer by clicking on the “Measure AB” checkbox. You may also click on the “Circle” checkbox to display a circle. Point F is the center and point G is a point on the circle. Both of these points can be moved to change the center’s location and the radius.