Exercise 24 Page 475

Let's begin by sketching the triangle with its angle bisectors.

Considering the Triangle-Angle-Bisector Theorem, let's find the lengths of each pair of segments one at a time.

Lengths of $\overline{BD}$ and $\overline{DC}$

We will first consider $BD$ and $DC.$

By the theorem, we can write a proportion for $BD$ and $DC.$ 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 $BD.$ Next, we will substitute the given values and $DC=13-BD$ into the proportion, so we can find the length of $\overline{BD}.$ Now, we can find $DC$ by substituting $BD\approx 3.8$ into $DC=13-BD.$

Lengths of $\overline{AE}$ and $\overline{EC}$

As our second pair of segments, we will think of $\overline{AE}$ and $\overline{EC}.$

Considering the Triangle-Angle-Bisector Theorem, let's write a proportion to find $AE$ and $EC.$ Next, we will rewrite $EC$ in terms of $AE$ using the fact that $AC=12.$ Now, we will first find $AE$ by substituting the value into the proportion. Now that we know $AE,$ we can find $EC$ as well.

Lengths of $\overline{AF}$ and $\overline{FB}$

Finally, we will find $AF$ and $FB.$

For $AF$ and $FB,$ we can write the following proportion. Next, we will again rewrite $FB$ in terms of $AF$ given that $AB=5.$ Now, let's find $AF.$ We have found that $AF=2.4.$ From here, we can also find $FB.$