# A triangle has corners at #(2 , 5 )#, ( 1, 3 )#, and #( 8, 1 )#. What are the endpoints and lengths of the triangle's perpendicular bisectors?

##### 1 Answer

I didn't finish, but here's some interesting stuff.

#### Explanation:

Why does this come up on "just asked" if it's two years old?

I recently gave a long rambling partial answer to one like this. I'll try to step up my game.

Let's first solve the following problem. We're given a triangle ABC where the midpoint of AB is the origin. We can label the points

I don't know how to do matrices here. Let's just consider the mapping

That's a dot product and a cross product, FYI. Let's look at our triangle under R:

In the transformed space, the perpendicular bisector of AB is

the y axis so the question has an obvious answer. The sign of ac+bd determines if the bisector hits A'C' (negative) or B'C' (positive).

The length of the bisector is the y intercept of the AC or BC as chosen. Let's work them out. The general line through

Line:

The y intercept is when

For

One of those is the length (or the signed length). Let's call it

The inverse transformation to R is

Let's map our y intercept back:

The denominator cancels the factor in the numerator of

That's pretty cool. I'm getting warnings that the answer is too long, so I'm just going to post this without finishing. Some problems are too long to do with a short answer.