Slope of Angle Bisector

Consider the line y = x and the line y = 0, both of which go through the origin. They form a 45° angle in the first quadrant. The bisector of this angle is therefore a line with y = mx for its equation. So what’s the value of m? It is tempting to think that it’s ½, but this is not the case.


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Recollection: a/b = c/d.

So:

2  =  1 – y
1 y

y2 = 1 – y
y2 + y = 1
y(1 + √2) = 1

y 1  =  1 (1 – √2)  =  1 – √2  = √2 – 1
1 + √2 (1 + √2) (1 – √2) 1 – 2

So y/1 = m, m = √2 – 1

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