Answer:
AB = 28 units
AC = 42 units
Step-by-step explanation:
I've attempted to recreate your description of the figure in the diagram below.
Angle Bisector Theorem
The angle bisector theorem suggests that any angle bisector inside of a triangle divides the side it intercepts in a ratio that is equal to the ratio of the two sides that enclose it.
Solving For x
In this question, for example, the following equality is true according to the angle bisector theorem:
[tex]\frac{AB}{AC} = \frac{BD}{DC}[/tex]
We can substitute AB = x - 7, AC = x + 7, BD = 8, CD = 12 into the equality and solve for x:
[tex]\frac{x-7}{x+7}=\frac{8}{12}\text{ //}\times12(x+7)\\\\12(x + 7)\frac{x - 7}{x + 7} = 12(x + 7)\frac8{12}\text{ // Simplify}\\\\12(x - 7)=8(x + 7)\\12x - 84 = 8x + 56\text{ //}-8x + 84\\12x - 8x = 56 + 84\\4x = 140\text{ //}\div4\\\\\boxed{x = 35}[/tex]
Finding AB and AC
Finally, to find AB and AC, we simply have to substitute x's value (35) back into AB and AC's lengths to work out their numerical length.
AB = x - 7 = 35 - 7 = 28 units
AC = x + 7 = 35 + 7 = 42 units
The lengths of AB and AC are 28 units and 42 units respectively.
