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Difference between revisions of "1982 AHSME Problems/Problem 10"

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== Problem 10 Solution ==
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== Problem ==
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In the adjoining diagram, <math>BO</math> bisects <math>\angle CBA</math>, <math>CO</math> bisects <math>\angle ACB</math>, and <math>MN</math> is parallel to <math>BC</math>.
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If <math>AB=12, BC=24</math>, and <math>AC=18</math>, then the perimeter of <math>\triangle AMN</math> is
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<asy>
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size(200);
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defaultpen(linewidth(0.7)+fontsize(10));
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pair B=origin, C=(24,0), A=intersectionpoints(Circle(B,12), Circle(C,18))[0], O=incenter(A,B,C), M=intersectionpoint(A--B, O--O+40*dir(180)), N=intersectionpoint(A--C, O--O+40*dir(0));
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draw(B--M--O--B--C--O--N--C^^N--A--M);
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label("$A$", A, dir(90));
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label("$B$", B, dir(O--B));
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label("$C$", C, dir(O--C));
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label("$M$", M, dir(90)*dir(B--A));
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label("$N$", N, dir(90)*dir(A--C));
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label("$O$", O, dir(90));</asy>
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<math>\textbf {(A)}\ 30 \qquad
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\textbf {(B)}\ 33 \qquad
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\textbf {(C)}\ 36 \qquad
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\textbf {(D)}\ 39 \qquad
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\textbf {(E)}\ 42  </math> 
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== Solution ==
  
 
Since <math>BO</math> and <math>CO</math> are angle bisectors of angles <math>B</math> and <math>C</math> respectively, <math>\angle MBO = \angle OBC</math> and similarly <math>\angle NCO = \angle OCB</math>. Because <math>MN</math> and <math>BC</math> are parallel, <math>\angle OBC = \angle MOB</math> and <math>\angle NOC = \angle OCB</math> by corresponding angles. This relation makes <math>\triangle MOB</math> and <math>\triangle NOC</math> isosceles. This makes <math>MB = MO</math> and <math>NO = NC</math>. <math>AM</math> + <math>MB</math> = 12, and <math>AN</math> + <math>NC</math> = 18. So, <math>AM</math> + <math>MO</math> = 12, and <math>AN</math> + <math>NO</math> = 18, and those are all of the lengths that make up <math>\triangle AMN</math>. Therefore, the perimeter of <math>\triangle AMN</math> is <math>12 + 18 = 30</math>. The answer is <math>\boxed{A}</math>.
 
Since <math>BO</math> and <math>CO</math> are angle bisectors of angles <math>B</math> and <math>C</math> respectively, <math>\angle MBO = \angle OBC</math> and similarly <math>\angle NCO = \angle OCB</math>. Because <math>MN</math> and <math>BC</math> are parallel, <math>\angle OBC = \angle MOB</math> and <math>\angle NOC = \angle OCB</math> by corresponding angles. This relation makes <math>\triangle MOB</math> and <math>\triangle NOC</math> isosceles. This makes <math>MB = MO</math> and <math>NO = NC</math>. <math>AM</math> + <math>MB</math> = 12, and <math>AN</math> + <math>NC</math> = 18. So, <math>AM</math> + <math>MO</math> = 12, and <math>AN</math> + <math>NO</math> = 18, and those are all of the lengths that make up <math>\triangle AMN</math>. Therefore, the perimeter of <math>\triangle AMN</math> is <math>12 + 18 = 30</math>. The answer is <math>\boxed{A}</math>.

Latest revision as of 12:31, 16 July 2024

Problem

In the adjoining diagram, $BO$ bisects $\angle CBA$, $CO$ bisects $\angle ACB$, and $MN$ is parallel to $BC$. If $AB=12, BC=24$, and $AC=18$, then the perimeter of $\triangle AMN$ is

[asy] size(200); defaultpen(linewidth(0.7)+fontsize(10)); pair B=origin, C=(24,0), A=intersectionpoints(Circle(B,12), Circle(C,18))[0], O=incenter(A,B,C), M=intersectionpoint(A--B, O--O+40*dir(180)), N=intersectionpoint(A--C, O--O+40*dir(0)); draw(B--M--O--B--C--O--N--C^^N--A--M); label("$A$", A, dir(90)); label("$B$", B, dir(O--B)); label("$C$", C, dir(O--C)); label("$M$", M, dir(90)*dir(B--A)); label("$N$", N, dir(90)*dir(A--C)); label("$O$", O, dir(90));[/asy]

$\textbf {(A)}\ 30 \qquad \textbf {(B)}\ 33 \qquad \textbf {(C)}\ 36 \qquad \textbf {(D)}\ 39 \qquad \textbf {(E)}\ 42$

Solution

Since $BO$ and $CO$ are angle bisectors of angles $B$ and $C$ respectively, $\angle MBO = \angle OBC$ and similarly $\angle NCO = \angle OCB$. Because $MN$ and $BC$ are parallel, $\angle OBC = \angle MOB$ and $\angle NOC = \angle OCB$ by corresponding angles. This relation makes $\triangle MOB$ and $\triangle NOC$ isosceles. This makes $MB = MO$ and $NO = NC$. $AM$ + $MB$ = 12, and $AN$ + $NC$ = 18. So, $AM$ + $MO$ = 12, and $AN$ + $NO$ = 18, and those are all of the lengths that make up $\triangle AMN$. Therefore, the perimeter of $\triangle AMN$ is $12 + 18 = 30$. The answer is $\boxed{A}$.

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