reflection point and mirror line?

Question

Find the reflection of the point A(3,-2) through the mirror line of y=1/2x-1. Please show every step. Thanks!

Answer 1

Here's one way to do it.

Find an equation for the line perpendicular to the mirror line and passing through A.

The slope of the mirror line is ½, so the slope of any line perpendicular to the mirror line is -2. So, using the point-slope form of the equation for a line, the line we're looking for has equation

y-(-2) = -2(x-3), or
y = -2x + 4

This line intersects the mirror line at the point (2,0) (I'm sure you can find this solution as well.) This intersection point is the midpoint of the line segment connecting the point A with its reflection. Let the coordinates of the reflection be (p,q). Then

½(p + 3) = 2
½(q + (-2)) = 0

Solve for p and q.