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.
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
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)
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?
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?
Thanks a lot, after following your suggestions I got (1,9).
no, that is not correct ...
my error was that I read your P(5,1) as (1,5)
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)
To find the reflected point Q, we can use the formula provided:
Set d = (x + (y - c) * a) / (1 + a^2)
This formula is derived from the concept of finding the distance from a point to a line. The numerator (x + (y - c) * a) represents the projection of the point (x, y) onto the line y = ax + c. The denominator (1 + a^2) is a scaling factor that depends on the slope of the line.
Once we have the value of d, we can use the following formulas to find the coordinates of Q:
x' = 2 * d - x
y' = 2 * d * a - y + 2c
These formulas calculate the coordinates of the reflected point by applying a reflection transformation. The x-coordinate of Q is obtained by taking twice the distance d and subtracting the original x-coordinate. The y-coordinate of Q is obtained by taking twice the distance d multiplied by the slope a, subtracting the original y-coordinate, and adding 2 times the y-intercept c.
In this specific problem, the line y = 1/2x + 2 is given, and we are asked to find the reflection of point P(5, 1) over this line. To find the coordinates of the reflected point Q, we substitute the values of P into the formulas:
x = 5
y = 1
a = 1/2
c = 2
Substituting these values into the formulas, we find:
d = (5 + (1 - 2) * (1/2)) / (1 + (1/2)^2) = (5 - 1/2) / (1 + 1/4) = (5 - 1/2) / (5/4) = (5 - 1/2) * (4/5) = (19/2) * (4/5) = 76/10 = 7.6
x' = 2 * 7.6 - 5 = 15.2 - 5 = 10.2
y' = 2 * 7.6 * (1/2) - 1 + 2 * 2 = 15.2 * (1/2) - 1 + 4 = 7.6 - 1 + 4 = 10.6
Therefore, the coordinates of the reflected point Q are approximately (10.2, 10.6).