Proof of the three perpendicular bisectors of the sides of a triangle are concurrent.
Perpendicular Bisector of a Triangle As we can see that all the perpendicular bisectors of a triangle meet at one single point which is called the circumcenter of . Perpendicular Bisectors of Triangle Meet at Point By definition of perpendicular bisector: From Triangle Side-Angle-Side Equality. The perpendicular bisector of a line segment can be constructed using a compass by drawing A triangle's three perpendicular bisectors meet (Casey , p.
And then we know that the CM is going to be equal to itself. And so this is a right angle. We have a leg, and we have a hypotenuse.
We know by the RSH postulate, we have a right angle. We have one corresponding leg that's congruent to the other corresponding leg on the other triangle. We have a hypotenuse that's congruent to the other hypotenuse, so that means that our two triangles are congruent. Well, if they're congruent, then their corresponding sides are going to be congruent. So that tells us that AM must be equal to BM because they're their corresponding sides.
So this side right over here is going to be congruent to that side. So this really is bisecting AB. So this line MC really is on the perpendicular bisector. And the whole reason why we're doing this is now we can do some interesting things with perpendicular bisectors and points that are equidistant from points and do them with triangles. So just to review, we found, hey if any point sits on a perpendicular bisector of a segment, it's equidistant from the endpoints of a segment, and we went the other way.
If any point is equidistant from the endpoints of a segment, it sits on the perpendicular bisector of that segment. So let's apply those ideas to a triangle now. So let me draw myself an arbitrary triangle. I'll try to draw it fairly large. So let's say that's a triangle of some kind. Let me give ourselves some labels to this triangle. That's point A, point B, and point C. You could call this triangle ABC.
Bisectors of Triangles
Now, let me just construct the perpendicular bisector of segment AB. So it's going to bisect it. So this distance is going to be equal to this distance, and it's going to be perpendicular. So it looks something like that.
And it will be perpendicular. Actually, let me draw this a little different because of the way I've drawn this triangle, it's making us get close to a special case, which we will actually talk about in the next video.
Let me draw this triangle a little bit differently. This one might be a little bit better. And we'll see what special case I was referring to. So this is going to be A.
This is going to be B. This is going to be C. Now, let me take this point right over here, which is the midpoint of A and B and draw the perpendicular bisector.
So the perpendicular bisector might look something like that. And I don't want it to make it necessarily intersect in C because that's not necessarily going to be the case.
But this is going to be a degree angle, and this length is equal to that length. And let me do the same thing for segment AC right over here.
- Perpendicular Bisector of a Triangle
- Perpendicular Bisector
- Circumcenter of a triangle
Let me take its midpoint, which if I just roughly draw it, it looks like it's right over there. And then let me draw its perpendicular bisector, so it would look something like this.
So this length right over here is equal to that length, and we see that they intersect at some point. Just for fun, let's call that point O.
And now there's some interesting properties of point O. We know that since O sits on AB's perpendicular bisector, we know that the distance from O to B is going to be the same as the distance from O to A.
That's what we proved in this first little proof over here. So we know that OA is going to be equal to OB. Well, that's kind of neat. But we also know that because of the intersection of this green perpendicular bisector and this yellow perpendicular bisector, we also know because it sits on the perpendicular bisector of AC that it's equidistant from A as it is to C.
So we know that OA is equal to OC. Now, this is interesting. OA is equal to OB. So we also know that OC must be equal to OB. OC must be equal to OB. Well, if a point is equidistant from two other points that sit on either end of a segment, then that point must sit on the perpendicular bisector of that segment. That's that second proof that we did right over here.
Perpendicular Bisector of a Triangle - Definition & Examples | [email protected]
So it must sit on the perpendicular bisector of BC. So if I draw the perpendicular bisector right over there, then this definitely lies on BC's perpendicular bisector.
And what's neat about this simple little proof that we've set up in this video is we've shown that there's a unique point in this triangle that is equidistant from all of the vertices of the triangle and it sits on the perpendicular bisectors of the three sides. Or another way to think of it, we've shown that the perpendicular bisectors, or the three sides, intersect at a unique point that is equidistant from the vertices.
And this unique point on a triangle has a special name. We call O a circumcenter. And because O is equidistant to the vertices, so this distance-- let me do this in a color I haven't used before. This distance right over here is equal to that distance right over there is equal to that distance over there. An important type of segment, ray, or line that can help us prove congruence is called an angle bisector. Understanding what angle bisectors are and how they affect triangle relationships is crucial as we continue our study of geometry.
Let's investigate different types of bisectors and the theorems that accompany them. Segment CD is the perpendicular bisector to segment AB. We derive two important theorems from the characteristics of perpendicular bisectors. We can use these theorems in our two-column geometric proofs, or we can just use them to help us in geometric computations.
Perpendicular Bisector Theorem If a point lies on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.
If a point is equidistant from the endpoints of a segment, then it lies on the perpendicular bisector of the segment. These theorems essentially just show that there exist a locus of points which form the perpendicular bisector that are equidistant from the endpoints of a given segment which meet at the midpoint of the segment at a right angle.
An illustration of this concept is shown below. Together, they form the perpendicular bisector of segment AB. The perpendicular bisectors of a triangle have a very special property. Let's investigate it right now. Circumcenter Theorem The perpendicular bisectors of the sides of a triangle intersect at a point called the circumcenter of the triangle, which is equidistant from the vertices of the triangle.
Point G is the circumcenter of? Angle Bisectors Now, we will study a geometric concept that will help us prove congruence between two angles. Any segment, ray, or line that divides an angle into two congruent angles is called an angle bisector. We will use the following angle bisector theorems to derive important information from relatively simple geometric figures.
Perpendicular Bisector -- from Wolfram MathWorld
Angle Bisector Theorem If a point lies on the bisector of an angle, then it is equidistant from the sides of the angle. If a point in the interior of an angle is equidistant from the sides of the angle, then it lies on the bisector of the angle. The points along ray AD are equidistant from either side of the angle. Together, they form a line that is the angle bisector. Similar to the perpendicular bisectors of a triangle, there is a common point at which the angle bisectors of a triangle meet.
Let's look at the corresponding theorem. Incenter Theorem The angle bisectors of a triangle intersect at a point called the incenter of the triangle, which is equidistant from the sides of the triangle. Point G is the incenter of? Summary While similar in many respects, it will be important to distinguish between perpendicular bisectors and angle bisectors.
We use perpendicular bisectors to create a right angle at the midpoint of a segment. Any point on the perpendicular bisector is equidistant from the endpoints of the given segment.Perpendicular Bisectors and the Circumcenter