2016 AMC 12A Problems/Problem 12

Solution

By the angle bisector theorem, $\frac{AB}{AE} = \frac{CB}{CE}$

$\frac{6}{AE} = \frac{7}{8 - AE}$ so $AE = \frac{48}{13}$

Similarly, $CD = 4$

Now, we use mass points.

Assign point $C$ a mass of $1$ .

Because $\frac{AE}{EC} = \frac{6}{7}, A$ will have a mass of $\frac{7}{6}$

Similarly, $B$ will have a mass of $\frac{4}{3}$

$mE = mA + mC = \frac{13}{6}$ .

Similarly, $mD = mC + mB = \frac{7}{3}$

The mass of $F$ is the sum of the masses of $E$ and $B$ .

$mF = mE + mB = \frac{7}{2}$

This can be checked with $mD + mA$ , which is also $\frac{7}{2}$

So $\frac{AF}{AD} = \frac{mD}{mA} = \boxed{\textbf{(C)}\; 2 : 1}$