Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

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

m (→‎Solution 2: The radius of a circle cannot equal to 0. Although this solution gives the right answer, it uses the wrong approach ...)
(→‎Solution 1: Some subtle points of this solution are missing. I will rewrite this solution a bit. Credits are retained to pieater314159.)
Line 5: Line 5:
 
<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>
  
−
== Solution 1 ==
+
== Solution ==
  
−
Let the triangle have coordinates <math>(0,0),(12,0),(0,5).</math> Then the coordinates of the incenter and circumcenter are <math>(2,2)</math> and <math>(6,2.5),</math> respectively. If we let <math>M=(x,x),</math> then <math>x</math> satisfies
+
~pieater314159 ~MRENTHUSIASM
−
<cmath>\sqrt{(2.5-x)^2+(6-x)^2}+x=6.5</cmath>
 
−
<cmath>2.5^2-5x+x^2+6^2-12x+x^2=6.5^2-13x+x^2</cmath>
 
−
<cmath>x^2=(5+12-13)x</cmath><cmath>x\neq 0\implies x=4.</cmath>Now the area of our triangle can be calculated with the Shoelace Theorem. The answer turns out to be <math>\boxed{\textbf{E}.}</math>
 
  
 
==See Also==
 
==See Also==

Revision as of 10:01, 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 $\triangle{MOI}$?

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

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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2018_AMC_12B_Problems/Problem_21&oldid=163802"