Unpacking the Distance Formula

Lesson 12

Unpacking the Distance Formula

by

Mr. White

Foundations of Mathematics Instuctor

Southern California

What is the Distance Formula?

The Distance Formula is derived from the Pythagorean Theorem, and is studied in Algebra and in the beginning of Plane Geometry. What the formula attempts to do is to allow students to calculate the distance between two points that line on an oblique line within a coordinate plane. Most teachers and books do not stress the oblique line as part of their work to teach the Distance Formula. But, to me, why do we need the formula if not for oblique lines? Anyone can count the distance between two points in a coordinate plane if they have access to a coordinate plane to view the vertical or original line, true? We need the Distance formula only when the line segment being measured is neither horizontal or vertical.

One of the confusing parts of the distance formula for students is the formula itself. The formula appears below:

Ö(x₂-x₁)²+ (y₂-y₁)²

What Do the Parts of the Distance Formula Really Mean?

The Distance Formula means nothing if we do not understand its components and how they are used in the formula. We must remember that we are evaluating the distance between two points, (x₂, y₂) and (x₁, y₁). The first part of the formula, (x₂-x₁)² allows us to subtract the two x coordinates of the two ordered pair we are using in the equation. The second part of the formula,(y₂-y₂)² , allows us to subtract the two y- coordinates of this same ordered pair. But why do we perform a squaring operation when we subtract the x's and the y's? The squaring is necessary because we want the difference between the two x- and the two y- coordinates to be positive. Distances can never be negative. When we add (x₂-x₁)²+ (y₂-y₁)², we are combining the value of the squared differences of the x's with the value of the squared differences of the y's. These squared distances are positive. We then take the quare root of this sum to obtain the true distance. Taking the square root of this squared number gives us the root( or the true value) for the distance between the two points.

Using the Distanceformulaa in an Example

Let us now use the paragraph above to guide us through a problem using the distance formula.

Given two points, A (0 , 0) and B ( 4, 3 ), what is the distance between the two points?
We first set up our equation:
Ö(4-0)²+(3-0)²
We solve that part of the equation involving the x- coordinates. 4-0 = 4. 4² = 16.
We then solve that part of the equation involving the y-coordinates. 3 - 0 = 3. 3² = 9.
We now have Ö(16 + 9) . This equals Ö(25). From Algebra we know that the square root of any positive number a is ±a. Since distance can never be negative, we chose the positive square root, 5. Therefore, the distance from point A to point B is 5 units.

Summary and Useful Web Links

All of the problems you will encounter using the Distance Forumla involve math facts that are described above. Use the links to the web pages below for additional leaopportunitiesnities.

http://whyslopes.com/Calculus-Introduction/distance_formula_and_midpoint_ca.html

http://www.purplemath.com/modules/distform.htm

The Midpoint Forumla Made Easy

Lesson 11 - Understanding the midpoint formula.
by
Mr. White
Foundationsl of Mathematics Instructor
Southern California

Vocabulary -

Midpoint
Coordinate/ ordered pair
X-
Y-
Line Segment
Endpoints
Average
Coordinate Geometry

The Midpoint Formula is used in coordinate geometry to allow us to determine the middle coordinate of a line segment. This point is also called the line bisector. It is as if you were measuring a line segment with a ruler. The middle of the line segment would be the midpoint.

For example, given two points, A (0,0) and B (4,3), what is the midpoint of the line that connects the two points, line segment AB? We could draw the line on a coordinate graph, but that is not necessary. We can use the Midpoint Formula, instead. This formula requires you to add the two X coordinates and average them. Then, add the two y coordinates, and average them. Those two answers are put together in the form of an ordered pair, which is the location of the line segment's middle point.

A (0, 0) B ( 4 , 4).

Calculate the midpoint of segment AB:

(4 + 0) /2 , (4 + 0)/2

This leads to the next step:

4/2, 4/2


which is the midpoint,

( 2, 2 )


Try another example.

Given the two points, R (1,2) and P (6, 5), what is the midpoint of the line segment, PR?

Solution: Add the x- coordinates, and divide by two, for the x-coordinate of the midpoint. Add the y- coordinates, and divide by two, for the y-coordinate of the midpoint.

Calculate the midpoint:

(1 + 7) / 2 , (2 + 6) / 2


This leads to the next step:

8/2, 8/2

or

(4, 4)

For more examples, go to these websites:

http://www.purplemath.com/modules/midpoint.htm

http://cs.selu.edu/~rbyrd/math/midpoint/

http://teacherschoice.com.au/Maths_Library/Analytical%20Geometry/Alg_16.htm

http://www.ronblond.com/M10/mid/index.html (A very interesting applet)