Angle Bisectors and Opposite Side Ratios - Concept
When an angle bisector is drawn in a triangle, the ratio of the opposite sides forming the bisected angle is equal to the ratio of the segments formed by bisector intersecting the opposite side. This ratio applies to all types of triangles and for an angle bisector drawn from any angle.
In this problem, all we have is a triangle and an angle bisector and we're being asked to find length x.
So let's go back to what we know. We know that if we have an angle bisector, it will divide the opposite side proportionally. That is we could say that a goes to x at that ratio is going to be equal to the ratio of b:y or you could also say that the ratio of a:b is the same as x:y.
So let's go back to our problem and apply what we know. We can say that the ratio of 6:x and we make that 6 a little bit darker, is equal to the ratio of 12:8. So we could cross multiply and divide here or you could see that to go from 12 to 6, we divide by 2, s to go from 8 to x we also need to divide by 2. Which means x must be 4.
So whenever you see an angle bisector, remember that it will divide the opposite side proportionally to the sides of the angle that you're bisecting.