Points, Lines, Angles, and Planes
Distance and Midpoint Formulas
Slapping coordinates on a line makes any line look like the number line. Just the same, this system of coordinates can make any plane look like the x-y plane from algebra.
Since lines only require one coordinate, they're one-dimensional. Planes require two, so they're two-dimensional. If we kick it up one more notch, we can put points in space (three-dimensional space, not the Final Frontier).
Finally, if we already know the exact point we're talking about, there's no need to add any coordinates to specify anything. That's why points are called zero-dimensional.
Going back to two dimensions, we can use coordinates to find the distance between points using the distance formula:
If M = (3, 4) and N = (5, -2), find the length of the segment MN.
As always, it's a good idea to sketch the problem first.
Since the length of MN is just the distance between the endpoints, we can use the distance formula.
Plugging this into a calculator, we get approximately 6.32 units, which looks about right on the sketch.
There's also a three-dimensional version of the distance formula (no 3D glasses required). It looks a lot like the two-dimensional version but with one addition.
You might be wondering why computing one-dimensional distance looks so different from these more complicated distance formulas in higher dimensions. Well, if we follow the pattern with the square root and everything using only one coordinate, we get:
d = |x2 – x1|
The square root cancels out the exponent, but we have to put absolute values since square roots, just like distances, are always positive. Basically, they're the same concepts extended to multiple dimensions.
The distance formula is really just an adaptation of the infamous Pythagorean theorem: a2 + b2 = c2. We'll talk about this more when we get into right triangles. We just said it because Greek influence extends even to Descartes's work. Geometry really is all about the Greeks, isn't it.
Coordinates also make it easy to find the midpoint of a segment. Finding the number in the middle of two others means finding their average (the average of a and b is (a + b) ÷ 2). Luckily, midpoint works the same way. Average each of the coordinates of the endpoints, and you've got a midpoint. Here's a formula, if you prefer to remember those:
That formula is for two dimensions, but if you're working in three dimensions (or somehow transcend into yet higher dimensional realms), follow the same pattern.
PQ has length 10 and has M as a midpoint. If P has coordinates (2, 4) and M has coordinates (5, 8), what are the coordinates of point Q?
It is tempting to plug (2, 4) and (5, 8) into the midpoint formula, but what would that give us? We'd just get the midpoint of P and M, which doesn't answer the question at all.
Instead, we are given a midpoint and need to solve for an endpoint. Let's just call the coordinates of Q (x, y) for now. We can still use the midpoint formula, but not in the same way we'd want to.
This looks like a single equation, but it's actually two equations disguised as one. We know that the first coordinates have to be equal, and also that the second coordinates are equal. In other words, we have these two equations.
Solving for x and y (or just guessing and checking) yields (x, y) = (8, 12), and we're done.
Hold up. We never used the length of PQ. Does that mean our answer is wrong? Not necessarily. Many times in geometry, we're given extra information to try to throw us off. Sometimes it's completely irrelevant ("If 1 + x = 83 and an orangutan's arms can extend up to 7 feet, find the value of x"). Sometimes it's actually pertinent to the question, but not needed to solve the problem. When that happens, we can use this extra information to check our answer.
In this case, we can check and see if the distance between P and Q is 10. If it is, then we have more proof (not proofs again!) that our answer is right. If we're wrong, we may want to go back and check our work.
PQ = 10
We're right, so there's no need to worry.