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## Geometry (all content)

### Unit 16: Lesson 4

Constructing a line tangent to a circle# Geometric constructions: circle tangent

CCSS.Math:

Sal constructs a line tangent to a circle using compass and straightedge.

## Video transcript

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.