Univ. of Wisconsin
J.D. Univ. of Wisconsin Law school
Brian was a geometry teacher through the Teach for America program and started the geometry program at his school
The shortest distance between a point not on a line and a line is along the perpendicular to the line. Constructing a perpendicular to a line uses the same process as constructing the perpendicular bisector of a line segment, but with one additional step. The first step is to swing an arc from the point and intersect the line in two places, which creates a segment that can be bisected.
If I showed you a line l and a point p that was not on that line, what's going to be the shortest distance from p to l? Well, let's say you didn't have any common sense and let's draw in a couple different lines. Let's say I said "Mr. McCall, shortest distance from p to l is along that line right there."
Well, just to give an idea, let's try measuring this and that distance is 9 inches. You're going to say "wait wait Mr. McCall, you got it all messed up." Shortest distance is at an angle like this. So if we measured that line segment, it's going to be about 12.5 inches. And then hopefully you'll say "wait a minute Mr. McCall. The shortest distance is just straight down. Otherwise known as a perpendicular line." So if we constructed a perpendicular line, and again I'm not constructing here because I didn't use a compass, then that's going to be our shortest distance. And just for giggles let's figure out what that distance is. That distance is 7.5 inches. So the shortest distance from p that's not on the line to a line is going to be a perpendicular.
But how are we going to construct that? Well, if we think about, if I erase this, if we think about ways that we know how to construct a perpendicular. What we're going to have to do is first, create two points on this line. Once we have those endpoints if we construct a perpendicular through them, and through point p, we're going to have a perpendicular bisector. So again the idea is to find a couple endpoints on this line that are the same distance from p and then to create the bisector that divides this line segment in half and is perpendicular to l.