A line with slope 1 bisects the angle from L sub1 to L sub2, where L sub2 has slope 2. What is the slope of?

Question

L sub1?..

Answer 1

If the bisecting line has slope 1, then the slopes of the two lines it bisects are reciprocals.

Since the slope of the given line is 2, the slope of the other line is 1/2.

The reciprocal rule only works when the slope of the bisecting line is 1 and the other lines are neither horizontal nor vertical.
______________

If you don't know the shortcut, use the angle subtraction formula for tangents. The slope of a line is the tangent of its angle.

Let the other angle be α.

Then the slope of the other line is tan(α).

And we have:

arctan(2) - arctan(1) = arctan(1) - α
arctan(2) - π/4 = π/4 - α

Take the tangent of both sides.

tan[arctan(2) - π/4] = tan[π/4 - α]

Apply the tangent angle subtraction formula.

[tan(arctan(2)) - tan(π/4)] / [1 + tan(arctan(2))*tan(π/4)]
= [tan(π/4) - tan(α)] / [1 + tan(π/4)*tan(α)]

(2 - 1) / (1 + 2*1) = [1 - tan(α)] / [1 + 1*tan(α)]

1/3 = [1 - tan(α)] / [1 + tan(α)]

1 + tan(α) = 3[1 - tan(α)] = 3 - 3tan(α)
4tan(α) = 2
tan(α) = 1/2

The slope of the other line is 1/2.

Answer 2

Use tangents.

L2 has slope 2, which means its angle from the x-axis is
atan(2) = 63.4349° ("atan" is arctangent, also known as inverse tangent)

The line with slope 1 has an angle from the x-axis of
atan(1) = 45°

So the angle L1 makes with the x-axis is 45°-(63.4349-45)° = 26.5651°. The tangent of this is the slope:
tan(26.5651°) = 0.50 or 1/2. (corrected an earlier mistake)