A few days ago I posted a “Friendly Formula” for the Midpoint Formula.
Today I am presenting a Friendly Formula for the Distance Formula, an important formula in Algebra 1 courses.
First I’m going to present the Friendly Formula for the Distance Formula and demonstrate how to use it. Then I’ll explain why it makes sense.
Buckle your seatbelts ’cause here it is: the distance between any two points on the coordinate plane is simply the SQUARE ROOT of … (the x-distance squared) plus (the y-distance squared).
And here’s an example of how easy it can be to use this formula. Suppose you want the distance between the points (2, 5) and (4, 9).
First figure out how the distance between the x-coordinates, 2 and 4. Well, 4 – 2 = 2, so the x-distance = 2. Now square that x-distance: 2 squared = 4
Next find the distance between the y-coordinates, 5 and 9: Well, 9 – 5 = 4, so the y-distance = 4. Now square that y-distance: 4 squared = 16
Next add the two squared values you just got: 4 + 16 = 20
Finally take the square root of that sum: square root of 20 = root 20.
That final value, root 20, is the distance between the two points.
Now we get to the question of WHY this Friendly Formula makes sense. I will explain that in my next post.
HINT: The Distance Formula is based on the Pythagorean Theorem. See if you can spot the connection.
EXTRA HINT: Make a coordinate plane. Plot the two points I used in this example, and construct a right triangle in which the line connecting these two points is the hypotenuse. If you can figure this out, the “Aha!” moment is a glorious event!
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In algebra we have many formulas to learn. But one problem is that those formulas are often hard to memorize. They are written with variables, and the variables frequently have subscripts, and the truth is that a lot of us don’t really understand what the formulas are saying or how they work. So of course that makes formulas difficult to memorize.
Enter the concept of “friendly formulas.” Friendly formulas are the very same formulas but written in a way that you can understand and therefore memorize much more easily. It’s an idea I have come up with through my many years of algebra tutoring, and idea is included in my Algebra Survival Guide, available through Amazon.com
In this post I describe the “friendly formula” for the midpoint formula.
So as a refresher, what is the midpoint formula all about?
Basically, it lets you find the midpoint of any line segment on the coordinate plane. Think of it this way. There’s some line segment on the coordinate plane called segment AB. That means that it has an endpoint at point A, another at point B. We are given the coordinate of points A and B. We want to find the coordinates of the point right in the middle of points A and B.
Now let’s make this idea easy. Suppose we focus only on the x-coordinates. Suppose the x-coordinate of point A is 2, and the x-coordinate of point B is 6. Ask yourself: what x-coordinate is perfectly in the middle of coordinates 2 and 6? It’s just like asking: what number is right in the middle of 2 and 6 on the number line? Well, wouldn’t that be 4, since 4 is two more than 2 and two less than 6? And indeed it is 4.
But notice that there’s another way to get 4, given the coordinates 2 and 6. We also could have just added 2 and 6 to get 8, and then divided 8 by 2, since 8 ÷ 2 = 4. In other words, we could have TAKEN the AVERAGE of the two x-coordinates, since taking an average of two numbers is adding them and dividing by two.
Could the midpoint formula actually be as easy as taking averages?!
Before we say yes, let’s test this idea for more complicated situations. We just saw that it works when both coordinates are positive. But suppose one coordinate is positive, the other negative. Let’s let one coordinate be – 2, while the other is + 4. What number is right between those two coordinates on the number line? Well, the numbers are 6 apart, right? And half of 6 is 3, so we could just add 3 to – 2, and get + 1 as the point in between them. And we see that + 1 is three away from both – 2 and 4. But could we also get + 1 by averaging -2 and 4? Let’s try: (- 2 + 4) / 2 = 2 / 2 = + 1. Averaging works again.
And finally, what about the case where both coordinates are negative? Suppose one coordinate is – 2, the other – 8. What number is right between those two numbers on the number line? Well, these numbers are also 6 apart, right? And half of 6 is 3, so we could just add 3 to – 8, and get – 5 as the middle. And we see that – 5 is three away from both – 8 and – 2. But can we also get – 5 by averaging – 8 and – 2? Let’s try: (- 8 + – 2) / 2 = – 10 / 2 = -5. Averaging worked here too!
Since the averaging process works for all three cases, this approach does works always, and in fact it is how the midpoint formula works.
The midpoint formula basically just averages the x-coordinates to get the x-coordinate of the midpoint. Then it averages the y-coordinates to get the y-coordinate of the midpoint.
So here is the “friendly formula” for the midpoint of any segment on the coordinate plane: Given a segment whose x- and y-coordinates are known,
MIDPOINT = (AVERAGE of x-coordinates, AVERAGEof y-coordinates)
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