If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

# Geometric constructions: circle tangent

CCSS.Math:

## Video transcript

you'd be amazed with the things that you can draw or construct if you have a straight edge and a compass and a straight edge is literally just something that that's a straight edge that allows you to draw straight lines like a ruler and a compass is something that allows you to draw circles centered where you want them to be centered of different radii and your typical compass it'd be like a metal thing it has a pin on one end and it's kind of shaped like an angle and then you have a pencil at the other end now 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 now I can 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 so I can move it around and I can draw this is equivalent to having a straightedge and so using these tools I want to construct a line going through P that is tangent to the circle now it might be tempting I could just let me just try to draw a line maybe it looks something like this remember a tangent a tangent line will touch the circle exactly one point and that point since it's going through B should be the point P another way to think about a tangent line is it's going to be perpendicular to the radius between that point and the center now what I just drew actually 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 at exactly one point right over there so what we're going to do is use our virtual compass and our virtual straightedge to try to do a more precise drawing so let's let's do that so the first thing I'm going to do is I'm going to set up P as the midpoint of a line where the center of the circle is one other end of the line and the way I can do that let me add a compass here and let me construct a circle that has the same okay so I've set up my compass to have the same radius as my original circle but now let me move it over right over here so now it's centered at P well why is this why is this useful well now a diameter of this new circle is going to be a segment that is centered at P so I'm going to have a segment where P is the midpoint and then up my R it my center of my original Circle is going to be one of the end points so let's do that so I'm going to add a straight edge so make that one of the end points and I'm going to go to go through P all the way to the other side of my new circle now what was the hole what's the whole point of me we'd try to do it as well as I can what's the whole point of me doing that well now I've made P the midpoint of a segment so if I can construct a perpendicular bisector to of this segment it will go through P because P is the midpoint and then that thing is going to be it's going to be exactly perpendicular to the radius because the radius the original actually for the original radius let me let me be careful here the original the original radius is part of is part of this segment so let's see how I could do this so what I could do is I'm going to add I'm going to draw another circle I'm going to Center this one at the original at the original circle and I'm going to make it have a different radius maybe a radius something like that and now let me get I'm going to construct another circle of this larger size but I'm going to Center it at this point right over here and I think you'll see quickly what this will accomplish so I'm going to construct another circle of that same larger radius of that same larger radius just like that and now I'm going to move it over here over here so what's interesting about the intersection of these two larger circles well this point right over here is equidistant to this this end of the segment and to this end of the segment cuz remember these two larger circles have the same radius so if I'm sitting on both of them then I am that distance away from this point and I am sitting that distance away from this point so something that is equidistant from the two endpoints of a segment they're going to sit it's going to sit on the perpendicular bisector so this point is going to sit on the perpendicular bisector and at this point is going to be sit on the perpendicular bisector so now we can draw a perpendicular bisector we can go from this point right over here the intersection of our two larger circles this point that is equidistant from my two centers of the large circle to this point that is equidistant to the two centers of the large circle and once again it's equidistant to this two centers of the large circle but those points are also the endpoints of this segment so these two points are on the perpendicular bisector you just need two points for line so I've just constructed a perpendicular bisector to P and it's perpendicular once again to the radius of our original the radius from the center to P of our original circle now that is going to be a tangent line because this right if you're if we go only through P or we go through P and we are exactly perpendicular to the radius from P to the center then this line that we've just constructed is actually tangent so anyway it might seem like a lot of work to do all of this you know I could have started just eyeballing it but when we do it like this we can feel really really really really really good that we're being precise imagine if you were trying to do this in a larger scale or if you're trying to engineer some very precise instrument you would want to do it this way you're going to draw a very precise drawing maybe an architectural drawing this could be an interesting way to to approach it and you know in times past before folks had things like computers and whatever else this this was a thing people actually did