# Analytic Geometry - Reflecting points over lines

posted by on .

Let P = (5,1), and let Q be the reflection of P over the line y = 1/2x + 2. Find the coordinates of Q.

I don't understand how to start? Should we draw perpendicular lines?
Analytic Geometry - Reflecting points over lines - Steve, Monday, March 4, 2013 at 12:15pm
Given (x,y) and a line y = ax + c we want the point (x', y') reflected on the line.

Set d:= (x + (y - c)*a)/(1 + a^2)

Then x' = 2*d - x

and y' = 2*d*a - y + 2c

This relies on the fact the the distance from (h,k) to the line ax+by+c = 0 is

|ah+bk+c|/√(a^2+b^2)

Question: How did you get this? I don't understand.

• Analytic Geometry - Reflecting points over lines - ,

A neat trick to see what Steve said is true...the distance from the line is the same.

Do this. TAke a piece of graph paper, plot the line, and the point. Then fold the paper along the line of symettry you drew, punch with a pin through the marked point so it goes through both sides of the paper. Unfold it. Notice the original point, and the new hole exactly the same distance from the line.
That is the "reflecton" of the point. Now reread Steves' solution

• Analytic Geometry - Reflecting points over lines - ,

As Steve told you, it uses the formula for the distance from a point to a line.

Here is another approach, perhaps it uses simpler algebra
let the reflected point be Q(a,b)
slope of given line is 1/2
slope of PQ = (b-5)/(a-1)
but they are perpendicular, so
(b-5)/(a-1) = -2
which after cross-multiplying and simplifying gives me
b = 7-2a
Clearly the midpoint of PQ must lie on the given line
midpoint is ( (a+1)/2 , (b+5)/2)
= ( (a+1)/2, (12-2a)/2 )

subbing into the equation
y = (1/2)x + 2
(12-2a)/2 = (1/2)((a+1)/2 + 2
times 2
12 - 2a = (a+1)/2 + 4
times 2 again
24 - 4a = a+1 + 8
-5a = -15
a = 3 , then b = 7-6 = 1

the point P is (3,1)

• Analytic Geometry - Reflecting points over lines - ,

THANKS A LOT EVERYONE but if the point Q is on 3,1 (I suppose that is what you meant) then i don't think its right since if it is reflected over the line 1/2x+2 then it would be on the other side, right?

• Analytic Geometry - Reflecting points over lines - ,

I really have to get new glasses, lol

but, ...
why don't you just change the numbers from (1,5) to (1,3) and follow the same steps?

• Analytic Geometry - Reflecting points over lines - ,

Thanks a lot, after following your suggestions I got (1,9).

• Analytic Geometry - Reflecting points over lines - ,

no, that is not correct ...

so let's go back and change my steps to :

let the reflected point be Q(a,b)
slope of given line is 1/2
slope of PQ = (b-1)/(a-5)
but they are perpendicular, so
(b-1)/(a-5) = -2
which after cross-multiplying and simplifying gives me
b = 11-2a
Clearly the midpoint of PQ must lie on the given line
midpoint is ( (a+5)/2 , (b+1)/2)
= ( (a+5)/2, (12-2a)/2 )

subbing into the equation
y = (1/2)x + 2
(12-2a)/2 = (1/2)((a+5)/2 + 2
times 2
12 - 2a = (a+5)/2 + 4
times 2 again
24 - 4a = a+5 + 8
-5a = -11
a = -11/-5 = 2.2 , then b = 11-4.4 = 6.6

the point P is (2.2, 6.6)

check:
slope PQ = (6.6 - 1)/(2.2 - 5) = 5.6/-2.8 = -2 , which is correct

distance for P(5,1) to line is
|5 - 2 + 4|/√(1^2+2^2) = 7/√5
distance for Q(2.2 , 6.6) to the line is
|2.2 - 13.2 + 4|/√5 = 7/√5

The point P(5,1) is reflected in the line
y = (1/2)x + 2 to the point Q(2.2 , 6.6) or Q(11/5 , 33/5)