# Find perpendicular bisector of a line segment where two faces meet

### Proof: The diagonals of a kite are perpendicular (video) | Khan Academy

A perpendicular bisector is a special kind of segment, ray, or line that The perpendicular bisectors of the sides of a triangle intersect at a point called the circumcenter of the triangle, which is Any segment, ray, or line that divides an angle into two congruent angles is called an angle bisector. Find the values of x and y. The perpendicular bisector of a line segment can be constructed using a compass by drawing circles centered at A and the rigid compass can be used to immediately draw the two arcs using any radius larger that half the length of AB A triangle's three perpendicular bisectors meet (Casey , p. Get your answers one. To find auxiliary points X, Y, Z on the bisectors of the line segments: To find intersection P of two bisectors AX and BY: \textit{The three perpendicular bisectors of the sides of a triangle meet in a point which is equally distant from the .

And then we have this angle bisector right over there. And we need to figure out just this part of the triangle, between this point, if we call this point A, and this point right over here. We need to find the length of AB right over here.

So once again, angle bisector theorem, the ratio of 5 to this, let me do this in a new color, the ratio of 5 to x is going to be equal to the ratio of 7 to this distance right over here. And what is that distance?

Well, if the whole thing is 10, and this is x, then this distance right over here is going to be 10 minus x. So the ratio of 5 to x is equal to 7 over 10 minus x. And we can cross multiply 5 times 10 minus x is 50 minus 5x. And then x times 7 is equal to 7x. Add 5x to both sides of this equation, you get 50 is equal to 12x. This implies straight, because they're both part of this larger triangle, they are the corresponding angles, so they're going to have the exact same measure.

Now it seems like we could do something pretty interesting with these two smaller triangles at the top left and the top right of this, looks like, a kite like figure. Because we have a side, two corresponding sides are congruent, two corresponding angles are congruent, and they have a side in common.

They have this side in common right over here. So let's first just establish that they have this side in common right over here. So I'll just write statement 6. We have CE, the measure or the length of that line, is equal to itself. Once again, this is just obvious. Obvious from diagram it's the same line. But now we can use that information. So we don't have three sides, we haven't proven to ourselves that this side is the same as this side, that DE has the same length as EB.

But we do have a side, an angle between the sides, and then another side. And so this looks pretty interesting for our side angle side postulate. And when I write the labels for the triangles, I'm making sure that I'm kind of putting the corresponding point. So I started at D, then went to C, then to E. So the corresponding I guess angle, or the corresponding point or vertex I could say, for this triangle right over here, is B.

So if I start with D, I start with B. C in the middle is the corresponding vertex for either of these triangles, so I put it in the middle.

### Perpendicular Bisector Theorem (Proof, Converse, Examples, & Video)

And then they both go to E. And that's just to make sure that we are specifying what's corresponding to what. And we know this, we know this is true, by side angle side. And the information we got from-- so we got this side is established that these two sides are congruent was from statement 1.

Then that these angles are congruent is from statement 5 right over here. And then statement 6 gave us the other side. Statement 6, just like that.

## Proof: The diagonals of a kite are perpendicular

And if we know that these triangles are congruent, that means that all of their corresponding angles are congruent. So we know, for example, that this angle right over here is going to be congruent to that angle over there. So let's write that down. And this comes straight from statement 7. Once again, they're congruent.

## Bisectors of Triangles

And then we also know-- we'll make statement 9. We also know that the measure of angle DEC-- or maybe we should just write it this way. We also note that the points at which angle bisectors meet, or the incenter of a triangle, is equidistant from the sides of the triangle.

Let's work on some exercises that will allow us to put what we've learned about perpendicular bisectors and angle bisectors to practice. Exercise 1 BC is the perpendicular bisector of AD. Find the value of x. The most important fact to notice is that BC is the perpendicular bisector of AD because, although it is just one statement, we can derive much information about the figure from it.

The fact that it is a perpendicular bisector implies that segment DB is equal to segment AB since it passes through the midpoint of segment AD.

### Perpendicular Bisector -- from Wolfram MathWorld

N is the circumcenter of? Find the values of x and y. We have In order to solve for y, we have to use the information given by the Circumcenter Theorem. This theorem states that the circumcenter is equidistant from the vertices of the triangle. Exercise 3 Find the value of x. The illustration shows that points A and B are equidistant from point L.

By the converse of the Angle Bisector Theorem, we know that L must lie on the angle bisector of?

- More about triangles
- Perpendicular Bisector
- Perpendicular Bisector Theorem

BYL, so we can solve for x as shown below: Exercise 4 QS is the angle bisector of? From the information we've been given, we know that? PQS is congruent to? SQR because QS bisects the whole angle,? We have been given the measure of the whole angle and the measure of?