SOLUTION: Find the equation of the line bisecting the acute angles formed between lines X+7y=6 and x-y=4.

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Question 946203: Find the equation of the line bisecting the acute angles formed between lines
X+7y=6 and x-y=4.
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
x%2B7y=6
7y=-x%2B6
y=-x%2F7%2B6%2F7
m%5B1%5D=-1%2F7
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x-y=4
y=x-4
m%5B2%5D=1
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To get the angle between the two lines using the slopes use,
tan%28theta%29=abs%28%28m%5B1%5D-m%5B2%5D%29%2F%281%2Bm%5B1%5Dm%5B2%5D%29%29
tan%28theta%29=abs%28%281-%28-1%2F7%29%29%2F%281-1%2F7%29%29=%288%2F7%29%2F%286%2F7%29=4%2F3
So then theta=53.13.
The line x-y=4 makes a 45 degree angle with the x-axis.
So starting from that slope and rotating clockwise by half of 53.13 will get us to the slope of the bisector.
45-53.13%2F2=18.43
m=tan%2818.43%29
m=1%2F3
Now find the point of intersection of the two lines,
y%2B4%2B7y=6
8y=2
y=1%2F4
Then,
x-1%2F4=16%2F4
x=17%2F4
Now use the point-slope form of a line,
y-1%2F4=%281%2F3%29%28x-17%2F4%29
highlight%28y=x%2F3-7%2F6%29
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