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?

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)

thanks a lot

To find the coordinates of Q, the reflection of point P over the line y = 1/2x + 2, you can follow these steps:

1. Find the slope of the given line: y = 1/2x + 2. Comparing it to the slope-intercept form y = mx + c, we see that the slope is 1/2.

2. Since the line is perpendicular to the given line, the slope of the perpendicular line is the negative reciprocal of the given line's slope. Thus, the slope of the perpendicular line is -2.

3. Now, with the slope of the perpendicular line and the coordinates of P, you can use the point-slope form of a line to write the equation of the perpendicular line passing through point P. The equation will have the form y - y1 = m(x - x1), where (x1, y1) are the coordinates of point P and m is the slope of the perpendicular line.

Substituting the values: y - 1 = -2(x - 5).

4. Simplify the equation: y - 1 = -2x + 10.

5. Rewrite the equation in the slope-intercept form y = mx + c: y = -2x + 11.

6. Set the equation of the perpendicular line equal to the equation of the given line to find their point of intersection. This is the point at which the reflection Q will lie.

-2x + 11 = 1/2x + 2.

7. Solve the equation for x:

-2x - 1/2x = 2 - 11,
-4x - x/2 = -9,
(-8x - x)/2 = -9,
-9x/2 = -9,
-9x = -18.

Divide both sides of the equation by -9 to isolate x: x = 2.

8. Now that you have the x-coordinate of the point of intersection, substitute it back into either equation (y = -2x + 11 or y = 1/2x + 2) to find the corresponding y-coordinate. Using y = -2x + 11:

y = -2(2) + 11,
y = -4 + 11,
y = 7.

Therefore, the coordinates of point Q, the reflection of P over the line y = 1/2x + 2, are (2, 7).