The red ray crosses the x-axis at x = 13. There are several ways to get this answer -- here's a fairly simple one using nothing but a little geometric construction reasoning and the equation of a line.
First, let's point out the most common mistake: the angle bisector of a triangle does not necessarily pass through the midpoint of the opposite side! In this case, in fact, the diagram seems to suggest that the red ray does not bisect the opposite side, and the diagram is right! However, there's one case where an angle bisector will bisect the opposite side, and that's the case of an isosceles triangle.
Now the length of the horizontal side of our triangle is 4, so we can
make an isosceles triangle by finding a point along the side between
(1,4) and (5,1) that is distance 4 from (1,4). The distance between
(1,4) and (5,1) is 5 (our given triangle is a 3-4-5 right triangle!), so
we want a point that is 4/5 of the way from (1,4) to (5,1). So:
The equation of the line containing the red ray is now easy, using "point-slope form":
Substituting in y = 0 will give us the x-intercept of this line: