The lines \(y = \frac{5}{12} x\) and \(y = \frac{4}{3} x\) are drawn in the coordinate plane. Find the slope of the line that bisects the angle between these lines.

Logic Jun 22, 2019

#1**+1 **

The lines y = (5/12)x and y = (4/3)x are drawn in the coordinate plane. Find the slope of the line that bisects the acute angle between these lines.

Here's one way, Logic.....but I don't know if it's the most elegant

Construct a circle centered at the origin with a radius of 1

The equation is

x^2 + y^2 = 1

We can find the x coordinate of the intersection of the first line and this circle,thusly :

x^2 + [ (5/12)x ]^2 = 1

x^2 + (25/144)x^2 =1

[144 + 25] / 144 x^2 = 1

[169] /144 x^2 =1

x^2 = 144/169

x = 12/13

And y = (5/12)(12/13) = 5/13

So ( 12/13, 5/13 ) is on the circle

Likewise...we can find the intersection of this circle with the second line :

x^2 +[ (4/3)x\^2 =1

x^2 + (16/9)x^2 = 1

[9 + 16 ] / 9 * x^2 =1

25/9 x^2 =1

x^2 = 9/25

x = 3/5

And y = (4/3)(3/5) = 4/5

So (3/5, 4/5) is on the circle

And a chord can be drawn on this circle with endpoints ( 12/13, 5/13) and (3/5, 4/5 )

And the midpoint of this chord is

( [12/13 + 3/5] / 2, [ 5/13 + 4/5 ] / 2 ) =

(99/130 , 77/130 )

And the line drawn from the origin to this point is the line we seek and it will have the slope

[77/130] / [99/130] = 77/99 = 7/9

CPhill Jun 22, 2019