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Difference between revisions of "2018 AMC 12B Problems/Problem 21"

(→‎Solution 1: Some subtle points of this solution are missing. I will rewrite this solution a bit. Credits are retained to pieater314159.)
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<math>\textbf{(A)}\ 5/2\qquad\textbf{(B)}\ 11/4\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 13/4\qquad\textbf{(E)}\ 7/2</math>
 
<math>\textbf{(A)}\ 5/2\qquad\textbf{(B)}\ 11/4\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 13/4\qquad\textbf{(E)}\ 7/2</math>
 +
 +
== Diagram ==
 +
<asy>
 +
/* Made by MRENTHUSIASM */
 +
size(250);
 +
 +
pair A, B, C, O, I, M;
 +
C = origin;
 +
A = (12,0);
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B = (0,5);
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C = origin;
 +
O = circumcenter(A,B,C);
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I = incenter(A,B,C);
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M = (4,4);
 +
fill(M--O--I--cycle,yellow);
 +
draw(A--B--C--cycle^^circumcircle(A,B,C)^^incircle(A,B,C)^^circle(M,4)^^M--O--I--cycle);
 +
dot("$A$",A,1.5*SE,linewidth(4));
 +
dot("$B$",B,1.5*NW,linewidth(4));
 +
dot("$C$",C,1.5*SW,linewidth(4));
 +
dot("$O$",O,1.5*dir((5,12)),linewidth(4));
 +
dot("$I$",I,1.5*S,linewidth(4));
 +
dot("$M$",M,1.5*N,linewidth(4));
 +
</asy>
 +
~MRENTHUSIASM
  
 
== Solution ==
 
== Solution ==

Revision as of 10:27, 20 October 2021

Problem

In $\triangle{ABC}$ with side lengths $AB = 13$, $AC = 12$, and $BC = 5$, let $O$ and $I$ denote the circumcenter and incenter, respectively. A circle with center $M$ is tangent to the legs $AC$ and $BC$ and to the circumcircle of $\triangle{ABC}$. What is the area of $\triangle{MOI}$?

$\textbf{(A)}\ 5/2\qquad\textbf{(B)}\ 11/4\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 13/4\qquad\textbf{(E)}\ 7/2$

Diagram

[asy] /* Made by MRENTHUSIASM */ size(250); pair A, B, C, O, I, M; C = origin; A = (12,0); B = (0,5); C = origin; O = circumcenter(A,B,C); I = incenter(A,B,C); M = (4,4); fill(M--O--I--cycle,yellow); draw(A--B--C--cycle^^circumcircle(A,B,C)^^incircle(A,B,C)^^circle(M,4)^^M--O--I--cycle); dot("$A$",A,1.5*SE,linewidth(4)); dot("$B$",B,1.5*NW,linewidth(4)); dot("$C$",C,1.5*SW,linewidth(4)); dot("$O$",O,1.5*dir((5,12)),linewidth(4)); dot("$I$",I,1.5*S,linewidth(4)); dot("$M$",M,1.5*N,linewidth(4)); [/asy] ~MRENTHUSIASM

Solution

~pieater314159 ~MRENTHUSIASM

See Also

2018 AMC 12B (Problems • Answer Key • Resources)
Preceded by
Problem 20 		Followed by
Problem 22
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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