Difference between revisions of "2011 AMC 12A Problems/Problem 13"

Revision as of 14:16, 19 July 2020

Problem

Triangle $ABC$ has side-lengths $AB = 12, BC = 24,$ and $AC = 18.$ The line through the incenter of $\triangle ABC$ parallel to $\overline{BC}$ intersects $\overline{AB}$ at $M$ and $\overline{AC}$ at $N.$ What is the perimeter of $\triangle AMN?$

$\textbf{(A)}\ 27 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 33 \qquad \textbf{(D)}\ 36 \qquad \textbf{(E)}\ 42$

Solution

Let $O$ be the incenter of $\triangle{ABC}$ . Because $\overline{MO} \parallel \overline{BC}$ and $\overline{BO}$ is the angle bisector of $\angle{ABC}$ , we have

$\angle{MBO} = \angle{CBO} = \angle{MOB} = \frac{1}{2}\angle{MBC}$

It then follows due to alternate interior angles and base angles of isosceles triangles that $MO = MB$ . Similarly, $NO = NC$ . The perimeter of $\triangle{AMN}$ then becomes $\begin{align*} AM + MN + NA &= AM + MO + NO + NA \\ &= AM + MB + NC + NA \\ &= AB + AC \\ &= 30 \rightarrow \boxed{(B)} \end{align*}$

Solution 2

Let $O$ be the incenter. $AO$ is the angle bisector of $\angle MAN$ . Let the angle bisector of $\angle BAC$ meets $BC$ at $P$ and the angle bisector of $\angle ABC$ meets $AC$ at $Q$ . By applying both angle bisector theorem and Menelaus' theorem,

$\frac{AO}{OP} \times \frac{BP}{BC} \times \frac{CQ}{QA} = 1$

$\frac{AO}{OP} \times \frac{12}{30} \times \frac{24}{12} = 1$

$\frac{AO}{OP}=\frac{5}{4}$

$\frac{AO}{AP}=\frac{5}{9}$

Perimeter of $\triangle AMN = \frac{12+24+18}{9} \times 5 = 30 \rightarrow \boxed{(B)}$

Solution 3

Like in other solutions, let O be the incenter of $\triangle ABC$ . Let $AO$ intersect $BC$ at $D$ . By the angle bisector theorem, $\frac{BD}{DC} = \frac{AB}{AC} = \frac{12}{18} = \frac{2}{3}$ . Since $BD+DC = 24$ , we have $\frac{BD}{24-BD} = \frac{2}{3}$ , so $3BD = 48 - 2BD$ , so $BD = \frac{48}{5}$ . By the angle bisector theorem on $\triangle ABD$ , we have $\frac{DO}{OA} = \frac{BD}{BA} = \frac{4}{5}$ , so $\frac{DA}{OA} = 1 + \frac{DO}{OA} = \frac{9}{5}$ , so $\frac{AO}{AD} = \frac{5}{9}$ . Because $\triangle AMN \sim \triangle ABC$ , the perimeter of $\triangle AMN$ must be $\frac{5}{9} (12 + 18 + 24) = 30$ , so our answer is $\boxed{\textbf{(B)}\ 30}$ .

Another way to find $\frac{AO}{OD}$ is to use mass points. Assign a mass of 24 to $A$ , a mass of 18 to $B$ , and a mass of 12 to $C$ . Then $D$ has mass 30, so $\frac{AO}{OD} = \frac{30}{24} = \frac{5}{4}$ .

Video Solution

https://www.youtube.com/watch?v=u23iWcqbJlE ~Shreyas S

@@ Line 23: / Line 23: @@
 \end{align*}
 </cmath>
+==Solution 2==
+Let <math> O </math> be the incenter. <math> AO </math> is the angle bisector of <math> \angle MAN </math>. Let the angle bisector of <math> \angle BAC </math> meets <math> BC </math> at <math> P </math> and the angle bisector of <math> \angle ABC </math> meets <math> AC </math> at <math> Q </math>. By applying both angle bisector theorem and Menelaus' theorem,
+<math>\frac{AO}{OP} \times \frac{BP}{BC} \times \frac{CQ}{QA} = 1 </math>
+<math>\frac{AO}{OP} \times \frac{12}{30} \times \frac{24}{12} = 1 </math>
+<math>\frac{AO}{OP}=\frac{5}{4} </math>
+<math>\frac{AO}{AP}=\frac{5}{9} </math>
+Perimeter of <math> \triangle AMN = \frac{12+24+18}{9} \times 5 = 30  \rightarrow \boxed{(B)}</math>
+==Solution 3==
+Like in other solutions, let O be the incenter of <math>\triangle ABC</math>. Let <math>AO</math> intersect <math>BC</math> at <math>D</math>. By the angle bisector theorem, <math>\frac{BD}{DC} = \frac{AB}{AC} = \frac{12}{18} = \frac{2}{3}</math>. Since <math>BD+DC = 24</math>, we have <math>\frac{BD}{24-BD} = \frac{2}{3}</math>, so <math>3BD = 48 - 2BD</math>, so <math>BD = \frac{48}{5}</math>. By the angle bisector theorem on <math>\triangle ABD</math>, we have <math>\frac{DO}{OA} = \frac{BD}{BA} = \frac{4}{5}</math>, so <math>\frac{DA}{OA} = 1 + \frac{DO}{OA} = \frac{9}{5}</math>, so <math>\frac{AO}{AD} = \frac{5}{9}</math>. Because <math>\triangle AMN \sim \triangle ABC</math>, the perimeter of <math>\triangle AMN</math> must be <math>\frac{5}{9} (12 + 18 + 24) = 30</math>, so our answer is <math>\boxed{\textbf{(B)}\ 30}</math>.
+Another way to find <math>\frac{AO}{OD}</math> is to use mass points. Assign a mass of 24 to <math>A</math>, a mass of 18 to <math>B</math>, and a mass of 12 to <math>C</math>. Then <math>D</math> has mass 30, so <math>\frac{AO}{OD} = \frac{30}{24} = \frac{5}{4}</math>.
 ==Video Solution==

2011 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 12	Followed by Problem 14
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