Answer to Question #5616 in Calculus for stanley

Question

find the slope of the line bisecting the angle from the line through (-2,2) and (2,0) to the line through (3,-15) and (10,9)

Answer 1

Let's find the equations of the lines.
Line through the points (-2,2) and (2,0):

(x+2)/(2-(-2)) = (y-2)/(0-2)
x = -2y+2
y = -0.5x+1

Line through the points (3,-15) and (10,9):

(x-3)/(10-3) = (y-(-15))/(9-(-15))
24x-72 = 7y+105
y = 24x/7 - 177/7.

So, the slope of the first line is k1 = -1/2 and the slope of the second one is 24/7.
The angles of this lines are
a1 = arctan(k1) = arctan(-1/2);
a2 = arctan(k2) = arctan(24/7).

There are two bisecting lines with the slopes
k3 = tan((a1+a2)/2) = tan((arctan(-1/2)+arctan(24/7))/2) ≈ 0.4367,
k4 = tan((a1-a2)/2) = tan((arctan(-1/2)-arctan(24/7))/2) ≈ -1.1982,
respectively.