Difference between revisions of "Mock Geometry AIME 2011 Problems/Problem 6"

(Solution)
m (Solution)
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The problem asks for the probability that point <math>P</math> is inside an equilateral triangle <math>A_1B_1C_1</math>. Let <math>x</math>, <math>y</math>, and <math>z</math> be the three distances from point <math>P</math> to each of the vertices, with <math>x</math> being the longest distance. Let's consider the case in which point <math>P</math> is actually on the line:
 
The problem asks for the probability that point <math>P</math> is inside an equilateral triangle <math>A_1B_1C_1</math>. Let <math>x</math>, <math>y</math>, and <math>z</math> be the three distances from point <math>P</math> to each of the vertices, with <math>x</math> being the longest distance. Let's consider the case in which point <math>P</math> is actually on the line:
 
<asy>
 
<asy>
unitsize(1cm);
+
unitsize(0.75cm);
 
draw((0,4*sqrt(3))--(8,4*sqrt(3)));
 
draw((0,4*sqrt(3))--(8,4*sqrt(3)));
 
draw((0,4*sqrt(3))--(4,0));
 
draw((0,4*sqrt(3))--(4,0));
Line 11: Line 11:
 
draw((6,4*sqrt(3))--(4,0));
 
draw((6,4*sqrt(3))--(4,0));
 
label("$x$",(5,2sqrt(3)),NNW);
 
label("$x$",(5,2sqrt(3)),NNW);
 
+
label("$y$", (3,0),N)
 +
label("$z$", (7,0),N)
 
</asy>
 
</asy>

Revision as of 20:33, 7 July 2019

Problem

Three points $A,B,C$ are chosen at random on a circle. The probability that there exists a point $P$ inside an equilateral triangle $A_1B_1C_1$ such that $PA_1=BC,PB_1=AC,PC_1=AB$ can be expressed in the form $\frac{m} {n},$ where $m,n$ are relatively prime positive integers. Find $m+n.$

Solution

The problem asks for the probability that point $P$ is inside an equilateral triangle $A_1B_1C_1$. Let $x$, $y$, and $z$ be the three distances from point $P$ to each of the vertices, with $x$ being the longest distance. Let's consider the case in which point $P$ is actually on the line:

unitsize(0.75cm);
draw((0,4*sqrt(3))--(8,4*sqrt(3)));
draw((0,4*sqrt(3))--(4,0));
draw((8,4*sqrt(3))--(4,0));
draw((6,4*sqrt(3))--(4,0));
label("$x$",(5,2sqrt(3)),NNW);
label("$y$", (3,0),N)
label("$z$", (7,0),N)
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