Is there a specific formula to reflect a point over ANY linear line?

Question

I know how to reflect (-5, 0) over y = 2x manually by algebra, but is there a formula for all? I remember there was. It had to do with absolute values and some stuff like that. Any you know of? There is, and I used to know that formula but I forgot.

Answer 1

Maybe you are thinking of the midpoint formula and the distance formula for checking your answer.

Let's illustrate this using the point (-5,0) and reflecting it over the line y=2x using algebra.

First, plot the point (-5,0) and graph the line y=2x on a piece of graph paper.

Now draw a line segment perpendicular to the line y=2x and going through this line down into the fourth quadrant. Somewhere on this line segment is the point representing the point (-5,0) reflected over the line y=2x. This point is in quadrant 4 and we have to find the coordinates of this point.

Now, this line segment is perpendicular to the line y=2x, so find the equation of this line segment using y=mx+b where m is the negative reciprocal of the slope, 2, of the line y=2x. The equation is y=(-1/2)x+b so far. Now, using the point (-5,0), which this line segment goes through, find the y-intercept "b".

0=(-1/2)(-5)+b, 0=(5/2)+b and b=(-5/2).

Now we use y=2x and y=(-1/2)x+(-5/2) and find the coordinates of the point of intersection of the line and the line segment. This point will be the midpoint of the line segment whose endpoints are (-5,0) and (x,y) where (x,y) are the coordinates of the reflected point over the line y=2x. The point (-5,0) and the reflected point (x,y) are equidistant from the line y=2x.

Solving...

y=2x

y=(-1/2)x+(-5/2)

2x=(-1/2)x+(-5/2)

(2 1/2)x=-(2 1/2)

x=-1 and therefore y=-2

(-1,-2) is the point of intersection of the perpendicular line segment and the line that you graphed. It is not the reflected point !! We are going to find that now using the midpoint formula.

(-5,0)--------------(-1,-2)--------------(x,y)

(-5+x)/2=-1 and (0+y)/2=-2

-5+x=-2 and x=3, 0+y=-4 and y=-4

The point (3,-4) is the point (-5,0) reflected over the line y=2x.

The distance between (-5,0) and (-1,-2) is 2SQRT(3)

The distance between (-1,-2) and (3,-4) is 2SQRT(3) also

The distance between (-5,0) and (3,-4) is 4SQRT(3) and half of that is 2SQRT(3)

The final answer is the point (3,-4) as stated above.