2017 AIME I Problems/Problem 15: Difference between revisions
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The area of the smallest equilateral triangle with one vertex on each of the sides of the right triangle with side lengths <math>2\sqrt{3},~5,</math> and <math>\sqrt{37},</math> as shown, is <math>\frac{m\sqrt{p}}{n},</math> where <math>m,~n,</math> and <math>p</math> are positive integers, <math>m</math> and <math>n</math> are relatively prime, and <math>p</math> is not divisible by the square of any prime. Find <math>m+n+p.</math> | The area of the smallest equilateral triangle with one vertex on each of the sides of the right triangle with side lengths <math>2\sqrt{3},~5,</math> and <math>\sqrt{37},</math> as shown, is <math>\frac{m\sqrt{p}}{n},</math> where <math>m,~n,</math> and <math>p</math> are positive integers, <math>m</math> and <math>n</math> are relatively prime, and <math>p</math> is not divisible by the square of any prime. Find <math>m+n+p.</math> | ||
[asy] | <math>[asy] | ||
size(5cm); | size(5cm); | ||
pair C=(0,0),B=(0,2*sqrt(3)),A=(5,0); | pair C=(0,0),B=(0,2*sqrt(3)),A=(5,0); | ||
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draw(A--B--C--A^^P--Q--R--P); | draw(A--B--C--A^^P--Q--R--P); | ||
dot(A--B--C--P--Q--R); | dot(A--B--C--P--Q--R); | ||
[/asy] | [/asy]</math> | ||
==Solution== | ==Solution== | ||
==See Also== | ==See Also== | ||
Revision as of 16:56, 8 March 2017
Problem 15
The area of the smallest equilateral triangle with one vertex on each of the sides of the right triangle with side lengths 
and 
as shown, is 
where 
and 
are positive integers, 
and 
are relatively prime, and is not divisible by the square of any prime. Find 
$[asy] size(5cm); pair C=(0,0),B=(0,2*sqrt(3)),A=(5,0); real t = .385, s = 3.5*t-1; pair R = A*t+B*(1-t), P=B*s; pair Q = dir(-60) * (R-P) + P; fill(P--Q--R--cycle,gray); draw(A--B--C--A^^P--Q--R--P); dot(A--B--C--P--Q--R); [/asy]$ (Error compiling LaTeX. Unknown error_msg)
