Difference between revisions of "2016 AMC 12A Problems/Problem 12"

Revision as of 02:26, 5 August 2021

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

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

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

Similarly, $CD = 4$ .

There are two ways to solve from here. First way:

Note that $DB = 7 - 4 = 3.$ By the angle bisector theorem on $\triangle ADB,$ $\frac{AF}{FD} = \frac{AB}{DB} = \frac{6}{3}.$ Thus the answer is $\boxed{\textbf{(C)}\; 2 : 1}$

Second way:

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

Solution 2

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 3

Draw the third angle bisector, and denote the point where this bisector intersects $AB$ as $P$ . Using angle bisector theorem, we see $AE=48/13 , EC=56/13, AP=16/5, PB=14/5$ . Applying Van Aubel's Theorem, $AF/FD=(48/13)/(56/13) + (16/5)/(14/5)=(6/7)+(8/7)=14/7=2/1$ , and so the answer is $\boxed{\textbf{(C)}\; 2 : 1}$ .

Solution 4

One only needs the angle bisector theorem, the segment addition postulate, and some simple algebra to solve this question.

The question asks for AF:DF. Apply the angle bisector theorem to $\triangle ABD$ to get the ratio $\frac {AF}{DF}$ :

$\frac {AF}{AB}$ = $\frac {DF}{BD}$ or, equivalently,

$\frac {AF}{DF}$ = $\frac {AB}{BD}$ .

AB is given. To find BD apply the angle bisector theorem to $\triangle ABC$ to get:

$\frac {BD}{AB}$ = $\frac {CD}{AC}$

---> $\frac {BD}{AB}$ = $\frac {BC - BD}{AC}$ since, by the segment addition postulate, BD + CD = BC

---> BD = $\frac {AB*BC}{AC + AB}$ .

Substituting this expression for BD into the proportion $\frac {AF}{DF}$ = $\frac {AB}{BD}$ yields:

$\frac {AF}{DF}$ = $\frac {AB}{BD}$ = $\frac {AC + AB}{BC}$ = $\frac {8 + 6}{7}$ = 2.

@@ Line 14: / Line 14: @@
 Similarly, <math>CD = 4</math>.
+There are two ways to solve from here.
+First way:
+Note that <math>DB = 7 - 4 = 3.</math> By the angle bisector theorem on <math>\triangle ADB,</math> <math>\frac{AF}{FD} = \frac{AB}{DB} = \frac{6}{3}.</math> Thus the answer is <math>\boxed{\textbf{(C)}\; 2 : 1}</math>
+Second way:
 Now, we use [[mass points]]. Assign point <math>C</math> a mass of <math>1</math>.

2016 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 11	Followed by Problem 13
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