Last week, in Fundamental Understanding of Mathematics LX, we demonstrated a fact about right triangles known as the Pythagorean theorem, and used that to figure out the length of a sloped line.
This week we are going to dig into the coordinate graph a bit, and develop last weeks concept into a more useful form for people who like algebra rather than geometry, and come up with what's commonly called the distance formula.
When we use the distance formula, we want to know how far it is from point A to point B. Mathematicians are always calculating the distance from point A to point B, because it's a lot less messy that calculating the distance from New York to Syracuse. If we know how far apart A and B are, then we also know the length of a straight line between A and B.
Let's start with a right triangle
Last week we found that the length of the black line, squared, was the sum of the length of the blue line squared plus the length of the red line squared. The classic geometric drawing of this fact is
which shows the three sides of a right triangle extended out to form squares, which is intended to help people remember the formula (area of the two smaller squares is the same as the area of the larger square.)
Today, though, we are going to look at this from a point A, point B perspective, and the two points we are interested in are the endpoints of the black line.
If we put the triangle on our coordinate graph, we can see the exact locations of the endpoints, in this case, A is at (2, 2) and B is at (10, 6). But a formula isn't much use if it only solves the problem for one case. A formula should be able to solve any case. To get to our abstract, any case scenario, we put the endpoints in unknown locations, and show the coordinates as simply x and y, for the x coordinate and the y coordinate of the points.
This creates a bit of a problem: if we only look at the coordinates, x and y, they look the same for both points. How can we tell them apart? One possible solution would be to give point B different coordinate names, perhaps w and z. But then we would lose the ability to see which was the x coordinate and which was the y coordinate. We'd have to take notes and constantly refer to our notes. That's tedious and boring!
The solution mathematicians adopted was to use subscripts, and to refer to the points not as A and B but simply as the beginning point (point 1) and the ending point (point 2). The x coordinate would get a subscript which shows whether it is point 1's x coordinate, or point 2's x coordinate:
x_1 or x_2 (notice the image below has "proper" subscripts, the ordinary typed version is shown here)
So our new, could be anywhere coordinates are:
Last week we discovered that if we squared the rise, and squared the run, and added these two amounts, they would be the same as the length of the line, squared. The run is the distance between these two points in the x direction.
Recall that both x coordinates represent, on a number line, the distance from the zero mark on the x axis. What you don't see is the number line, but, without the distracting grid marks, it would look something like this:
With the extra number line details filled in, it's clear that the distance we are looking for, the distance in the x direction between x_1 and x_2, is found by subtracting x_2 - x_1 .
The same thinking applies to the rise, or the difference between the height of y_1 and the height of y_2.
The rise is y_2 - y_1
Last week, to find the area of the smallest square (in that case, the rise was the smallest distance, and made the smallest square, as it does this week,) we simply multiplies rise x rise. This week, we can use our subtraction (y_2 - y_1) to substitute for the rise, since they are the same distance. Likewise, we can use (x_2 - x_1) to replace run in the calculation run x run.
So, to revisit last weeks work, we came up with
unknown length x unknown length = (run x run) + (rise x rise)
Another mathematical notation involves repeated multiplication. When we multiply a length by the same length, we say we square the length, since we show it by drawing a square, with the sides "length" long. We write these repeated multiplications using exponents, like this:
Replacing all our repeated multiplications with exponents, we get
Notice that when we multiply a number by itself, or square it, the square is always a positive amount. So even if our line sloped down instead of up, and rise was a negative number, rise squared would still be a positive area.
We can replace rise and run with their coordinate subtraction equivalents
At this point, we pull out the square root brackets, and take square roots. On one side, we have a square of area length squared, which has sides of length, so the square root of length squared is simply length.
On the other side, though, we have a calculated number, based on the rise and run, each squared to get the areas of the smaller squares, which are then added together to get the area of the large square with sides of "length". So, while we wave our hands to distract our reader, we pull out our handy calculator and discover that the square root of
The distance formula, then, is the length of a straight line drawn between the beginning point and the ending point, or
Have fun in the comments.