Sal constructs a line tangent to a circle using compass and straightedge.
You'd be amazed at the things you can draw or construct if you have a straightedge and a compass. A straightedge is literally just something that has a straight edge that allows you to draw straight lines, like a ruler. A compass is something that allows you to draw circles centered where you want them to be centered of different radii. Your typical compass would be a metal thing that has a pin on one end and is shaped like an angle and you have a pencil at the other end. I don't have real physical rulers and pencils in front of me but I have the virtual equivalent. I can say 'add a compass' and draw a circle. I can pick where I want to center it and I can change the radius, and I can draw a straight line segment and move it around. This is equivalent to having a straightedge. Using these tools, I want to construct a line going through P that is tangent to the circle. I'll draw a line that looks something like this. Remember a tangent line will touch the circle exactly at one point and that point, since it's going through b should be point P. Another way to think about a tangent line is that it's going to be perpendicular to the radius between that point and the center. What I just drew looks pretty good, but it's not so precise. I don't know if it's exactly perpendicular to the radius, I don't know if it's touching it exactly at one point right there. What we're going to do is use our virtual compass and our virtual straightedge to do a more precise drawing. Let's do that. The first thing I'm going to do is set up P as the mid-point of a line where the center of the circle is one other end of the line. The way I can do that-- let me 'add a compass' here. I'll construct a circle that has the same radius as my original circle. Now let me move it over so now it's centered at P. Why is this useful? Now a diameter of this new circle is going to be a segment that is centered at P. I'm going to have a segment where P is the mid-point and then the center of my original circle is going to be one of the end points. Let's do that. I'm going to add a straightedge and make that one of the end points and I'm going to go through P all the way to the other side of my new circle. What was the whole point of me doing that? Now I've made P the mid-point of a segment so if I can construct a perpendicular bisector of the segment it will go through P because P is the mid-point and that thing is going to be exactly perpendicular to the radius because the original radius is part of this segment. Let's see how I can do this. What I could do is-- I'm going to draw another circle. I'm going to center it at the original circle and make it have a different radius. Maybe a radius something like that. Now I'm going to construct another circle of this larger size but I'm going to center it at this point over here. I think you'll see quickly what this will accomplish. So I'll construct another circle of that same larger radius. Now I'm going to move it over here. So what's interesting about the intersection of these two larger circles? This point over here is equidistant to this end of the segment and to this end of the segment. Remember these two larger circles have the same radius. So if I'm sitting on both of them I am that distance away from this point and that distance away from this point. So something that is equidistant from the two end points of a segment is going to sit on the perpendicular bisector. So this point is going to sit on the perpendicular bisector and this point is going to be sitting on the perpendicular bisector. So now we can draw a perpendicular bisector. We can go from this point, the intersection of our two larger circles this point that is equidistant from the two centers of the large circle, to this point that is equidistant to the two centers of the large circle. And once again, it is equidistant to the two centers of the large circle, but those points are also the end points of this segment. So these two points are on the perpendicular bisector, you just need two points for a line. So I've just constructed a perpendicular bisector to P and it's perpendicular to the radius from the center to P of our original circle. Now, that is going to be a tangent line because if we go through P and we are exactly perpendicular to the radius from P to the center then this line we've just constructed is actually tangent. So, it might seem like a lot of work to do all of this, I could have started just eyeballing it but when we do it like this we can feel really good that we're being precise. Imagine if you were trying to do this on a larger scale, if you were trying to engineer some very precise instrument, you would want to do it this way. You would want to draw a very precise drawing, maybe an architectural drawing. This could be an interesting way to approach it. In times past before folks had things like computers, this was a thing people actually did.