Question

Let
\[
1 + 3 x + 4 y =0\\
-1 + 5 x + 12 y= 0
\]
Be the equation of two lines. Show that
the line
\[
y=\frac{1}{14} (-8 x-1)
\]
is a bisector of one of the two angles that the first two lines make at their intersection point.

Answer 1

Here is a graph of the two lines

Answer 2

i would use the dot product theorem to show the two smaller triangles are equal

Answer 3

the direction vector of each line is just the components of their slope

Answer 4

This would work. Try it

Answer 5

You mean smaller angles and not smaller triangles.

Answer 6

well, there is no triangle here. So the smaller angles where the angle bisector bisects

Answer 7

I was referring to your post
"i would use the dot product theorem to show the two smaller triangles are equal"

Answer 8

ahhhhhhhhh..... i meant angles XD

Answer 9

vector of the first equation is <4,-3>
middle : <14,-8>
third: <12,-5>

so angle formed by <4,-3> and <12,-5> should be equal to the angle formed by <14,-5> and <12,-5>

Answer 10

it meant <14,-8> (not <14,-5>)

Answer 11

I found a formula for the line that bisector line. It derives the equation from the fact that each point on the bisector line is equidistant from the other two lines.
http://www.ditutor.com/line/equation_bisector.html

Answer 12

that would certainly be another good way to approach this problem