# Symmetric of a point relative to a line

1. Dec 24, 2011

### Acut

1. The problem statement, all variables and given/known data
What is the easiest way of finding the symmetrical of a point relative to an arbitrary line?
(I was asked on an exam to find the symmetrical of a point relative to the line y = x, but that's rather trivial - just switch the coordinates. How can I do it for any arbitrary line ax + by = c?)

2. Relevant equations

3. The attempt at a solution
I found a way, but it's rather messy. Given the line s and the point P, find the line t that is perpendicular to s and passes through P. Calculate the distance d between P and s. Find the point on the line t that is also at a distance d from s.

This sounds awfully complicated and messy for me. Is there a quicker way?

2. Dec 25, 2011

### Staff: Mentor

That's the way I'd do it. I don't know of any other.

3. Dec 25, 2011

### Hitchens

For a line of the form Ax+By+C = 0, distance d from a point p is:

d = |Ax+By+C|/(A2+B2)1/2

[|...| is simply the absolute value meant to keep d>/=0]

4. Dec 25, 2011

### Acut

This is precisely the formula I would use in the description I have given. But it's rather
cumbersome to use it twice in a single problem. Is there a way around?

5. Dec 26, 2011

### Hitchens

Why would you use it twice? Using it once will provide you with the shortest distance from the line to the point.