Let's begin by sketching the triangle with its .
Considering the , let's find the lengths of each pair of segments one at a time.
Lengths of BD and DC
We will first consider BD and DC.
By the theorem, we can write a proportion for
BD and
DC.
ACAB=DCBD
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 , 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.
ACAB=DCBD
125=13−BDBD
5(13−BD)=12BD
65−5BD=12BD
65=17BD
1765=BD
BD=1765
BD=3.82352…
BD≈3.8
Now, we can find
DC by substituting
BD≈3.8 into
DC=13−BD.
Lengths of AE and EC
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.
BCAB=ECAE
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.
BCAB=ECAE
135=12−AEAE
5(12−AE)=13AE
60−5AE=13AE
60=18AE
1860=AE
310=AE
AE=310
AE=3.33333…
AE≈3.3
Now that we know
AE, we can find
EC as well.
Lengths of AF and FB
Finally, we will find AF and FB.
For
AF and
FB, we can write the following proportion.
BCAC=FBAF
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.
BCAC=FBAF
1312=5−AFAF
12(5−AF)=13AF
60−12AF=13AF
60=25AF
2560=AF
512=AF
AF=512
AF=2.4
We have found that
AF=2.4. From here, we can also find
FB.