Difference between revisions of "Mock AIME 4 2006-2007 Problems/Problem 11"
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Let <math>AD = CE = a</math>, and <math>EA = DB = b</math>. Note that we want to compute the ratio <math>\frac{a}{b}</math>. | Let <math>AD = CE = a</math>, and <math>EA = DB = b</math>. Note that we want to compute the ratio <math>\frac{a}{b}</math>. | ||
Latest revision as of 19:33, 9 February 2017
Problem
Let 
be an equilateral triangle. Two points 
and 
are chosen on 
and 
, respectively, such that 
. Let 
be the intersection of 
and 
. The area of is 13 and the area of 
is 3. If 
, where 
, 
, and 
are relatively prime positive integers, compute 
.
Solution
Let 
, and 
. Note that we want to compute the ratio 
.
Assign a mass of 
to point 
. This gives point 
a mass of 
and point a mass of 
. Thus, 
.
Since the ratio of areas of triangles that share an altitude is simply the ratio of their bases, we have that:
.
By the quadratic formula, we find that 
, so 
.
Thus, our final answer is 
.
