By the theorem, we can write a proportion for BD and DC.
AB/AC=BD/DC
We have already been given AB=5 and AC=12. In this case, to find BD and DC, we should identify a relation between them. Knowing that BC=13 and considering the Segment Addition Postulate, let's write DC in terms of BD.
BC=BD+DC
⇓
DC= 13-BD
Next, we will substitute the given values and DC=13-BD into the proportion, so we can find the length of BD.
As our second pair of segments, we will think of AE and EC.
Considering the Triangle-Angle-Bisector Theorem, let's write a proportion to find AE and EC.
AB/BC=AE/EC
Next, we will rewrite EC in terms of AE using the fact that AC=12.
AC=AE+EC
⇓
EC= 12-AE
Now, we will first find AE by substituting the value into the proportion.
For AF and FB, we can write the following proportion.
AC/BC=AF/FB
Next, we will again rewrite FB in terms of AF given that AB=5.
AB=AF+FB
⇓
FB= 5-AF
Now, let's find AF.
