Question 1158186: Let K = (5,12), L = (14,0), and M = (0,0). The line x+2y = 14 bisects angle MLK. Find equations for the bisectors of angles KML and MKL. Is the slope of segment MK twice the slope of the bisector through M? Should it have been? Show that the three lines concur at a point C. Does C have any special significance?
Answer by greenestamps(13200)   (Show Source):
You can put this solution on YOUR website!
Given K(5,12), L(14,0), and M(0,0), the lengths of the sides of triangle KLM are KM=13, LM=14, and KL=15.
An angle bisector of a triangle divides the opposite side into two parts in the same ratio as the ratio of the sides of the triangle that form the angle. So
the bisector of angle M divides side KL into two parts in the ratio 13:14;
the bisector of angle L divides side KM into two parts in the ratio 14:15;
the bisector of angle K divides side LM into two parts in the ratio 13:15.
(1) The equation of the bisector of angle L is given as x+2y=14. Use the preceding to verify that equation.
The x-coordinate of the point where the bisector of angle L intersects KM is
The y-coordinate of that point is 
Using the two points (14,0) and (70/29,168/29), the slope of that angle bisector is -1/2. That leads to the equation x+2y=14, as given in the statement of the problem.
That provides some reassurance that the method we are using is valid.
(2) Use the same process to find the equation of the bisector of angle M.
The endpoints of segment KL are (5,12) and (14,0). So
The x-coordinate of the point where the bisector of angle M intersects KL is 
The y-coordinate of that point is 
Using the points (0,0) and (28/3,56/9), the slope of the bisector is 2/3; then the equation is y = (2/3)x.
(3) Use the same process to find the equation of the bisector of angle K.
The bisector of angle K divides side ML into two parts in the ratio 13:15.
That makes the coordinates of the point where the bisector intersects ML (13/2,0).
The two points (5,12) and (13/2,0) then make the equation of that angle bisector y = -8x+52.
So the equations of the three angle bisectors are
x+2y=14
y=(2/3)x
y = -8x+52
The slope of KM is 12/5; the slope of the angle bisector of angle M is 2/3. The slope of KM is NOT twice the slope of the angle bisector. There is no reason it should be.
What should be true -- and is -- is that the slopes of KM and the bisector of angle M should satisfy the tangent double angle formula:
The three angle bisectors intersect at a single point. Relatively simple algebra shows the point of intersection is (6,4).
Here is a graph of the 3 angle bisectors:
All the points on an angle bisector are, by definition, the same distance from the two sides of the angle. So, in a triangle, the intersection of the three angle bisectors is the only point that is the same distance from all three sides of the triangle.
That point is the center of the inscribed circle of the triangle.
The formula for the distance from a point to a line can be used to show that the point (6,4) is indeed the same distance from KL, LM, and MK.
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