# Proving the equation of perpendicular bisector

• ArL
In summary, the perpendicular bisector of two points (x1, y1) and (x2, y2) can be represented by the general form Ax + By = C, where A= 2(x2-x1), B= 2(y2 -y1) and C= y2^2 - y1^2 + x2^2 - x1^2. This can be found using parametric equations, specifically D = 1/2 [x_{1} + x_{2}, y_{1} + y_{2}] + t[-{\Delta y},{\Delta x}] where t is a parameter.
ArL

## Homework Statement

My question is to :

Prove that the perpendicular bisector of the two points (x1, y1) and (x2, y2), in general form, is given by

Ax + By = C

where

A= 2(x2-x1), B= 2(y2 -y1) and C= y2^2 - y1^2 + x2^2 - x1^2

Help me with this proving things.

Thanks

## The Attempt at a Solution

Don't know where to start.

I have no idea.

Try to find the equation of the line between the two points to start off with.

I've done it.

However, I got some questions. (to check if I am right)

if it is (X2-X1)((X2+X1)*1/2)

is this transformed to

((X2-X1)(X2+X1))/2 ?

which is expended to (X2^2-X1^2)/2 ?

I am not good at those if there are fractions in it.

Thanks.

You can do it pretty quickly using parametric equations. You would have

$$D = 1/2 [x_{1} + x_{2}, y_{1} + y_{2}] + t[-{\Delta y},{\Delta x}]$$

And then you could easily find it.

Last edited:

## 1. What is the perpendicular bisector theorem?

The perpendicular bisector theorem states that if a point lies on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. In other words, the perpendicular bisector divides the segment into two equal parts.

## 2. How do you prove that a line is the perpendicular bisector of a segment?

To prove that a line is the perpendicular bisector of a segment, you can use the following steps:

1. Construct a perpendicular line through the midpoint of the segment.
2. Show that the two resulting segments are equal in length.
3. Use the definition of a perpendicular bisector theorem to show that the point on the perpendicular line is equidistant from the endpoints of the original segment.

## 3. Can you use the perpendicular bisector theorem to find the midpoint of a segment?

Yes, the perpendicular bisector theorem can be used to find the midpoint of a segment. The point where the perpendicular bisector intersects the segment is the midpoint.

## 4. Is the perpendicular bisector theorem only applicable to straight lines?

Yes, the perpendicular bisector theorem is only applicable to straight lines. It states that the perpendicular bisector of a straight line segment is a straight line itself.

## 5. How is the perpendicular bisector theorem used in real life?

The perpendicular bisector theorem is used in many real-life applications, such as construction, engineering, and navigation. It is also used in computer graphics and image processing to create symmetrical images and designs. Additionally, the theorem is used in the field of geometry and can be applied to solve various problems involving perpendicular lines and bisectors.

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