2016 AMC 12A Problems/Problem 12

Revision as of 20:08, 19 February 2016 by I dont do math (talk | contribs) (Solution 4)

Problem 12

In $\triangle ABC$, $AB = 6$, $BC = 7$, and $CA = 8$. Point $D$ lies on $\overline{BC}$, and $\overline{AD}$ bisects $\angle BAC$. Point $E$ lies on $\overline{AC}$, and $\overline{BE}$ bisects $\angle ABC$. The bisectors intersect at $F$. What is the ratio $AF$ : $FD$?

[asy] pair A = (0,0), B=(6,0), C=intersectionpoints(Circle(A,8),Circle(B,7))[0], F=incenter(A,B,C), D=extension(A,F,B,C),E=extension(B,F,A,C); draw(A--B--C--A--D^^B--E); label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,N); label("$D$",D,NE); label("$E$",E,NW); label("$F$",F,1.5*N); [/asy]

$\textbf{(A)}\ 3:2\qquad\textbf{(B)}\ 5:3\qquad\textbf{(C)}\ 2:1\qquad\textbf{(D)}\ 7:3\qquad\textbf{(E)}\ 5:2$

Solution 1

Applying the angle bisector theorem to $\triangle ABC$ with $\angle CAB$ being bisected by $AD$, we have

\[\frac{CD}{AC}=\frac{BD}{AB}.\]

Thus, we have

\[\frac{CD}{8}=\frac{BD}{6},\]

and cross multiplying and dividing by $2$ gives us

\[3\cdot CD=4\cdot BD.\]


Since $CD+BD=BC=7$, we can substitute $CD=7-BD$ into the former equation. Therefore, we get $3(7-BD)=4BD$, so $BD=3$.


Apply the angle bisector theorem again to $\triangle ABD$ with $\angle ABC$ being bisected. This gives us

\[\frac{AB}{AF}=\frac{BD}{FD},\]

and since $AB=6$ and $BD=3$, we have

\[\frac{6}{AF}=\frac{3}{FD}.\]

Cross multiplying and dividing by $3$ gives us

\[AF=2\cdot FD,\]

and dividing by $FD$ gives us

\[\frac{AF}{FD}=\frac{2}{1}.\]

Therefore,

\[AF:FD=\frac{AF}{FD}=\frac{2}{1}=\boxed{\textbf{(C)}\; 2 : 1}.\]

Solution 2

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 $1$.

$mC \cdot CD = mB \cdot DB$ , so $mB = \frac{4}{3}$

Similarly, $A$ will have a mass of $\frac{7}{6}$

$mD = mC + mB = 1 + \frac{4}{3} = \frac{7}{3}$

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

Solution 3

Denote $[\triangle{ABC}]$ as the area of triangle ABC and let $r$ be the inradius. Also, as above, use the angle bisector theorem to find that $BD = 3$. There are two ways to continue from here:

$1.$ Note that $F$ is the incenter. Then, $\frac{AF}{FD} = \frac{[\triangle{AFB}]}{[\triangle{BFD}]} = \frac{AB * \frac{r}{2}}{BD * \frac{r}{2}} = \frac{AB}{BD} =  \boxed{\textbf{(C)}\; 2 : 1}$

$2.$ Apply the angle bisector theorem on $\triangle{ABD}$ to get $\frac{AF}{FD} = \frac{AB}{BD} = \frac{6}{3} = \boxed{\textbf{(C)}\; 2 : 1}$

Solution 4

INDUCTIVE REASONING IS STILL REASONING

Tear a piece of your scrap paper and mark the length of $AF$ on it with your pencil. Do the same for $FD$. Clearly $AF$ is twice $FD$. Thus $\frac{AF}{FD} = \boxed{\textbf{(C)}\; 2 : 1}$

See Also

2016 AMC 12A (ProblemsAnswer KeyResources)
Preceded by
Problem 11
Followed by
Problem 13
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AMC 12 Problems and Solutions

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