2012 AMC 10B Problems/Problem 21
Problem 21
Four distinct points are arranged on a plane so that the segments connecting them have lengths ,
,
,
,
, and
. What is the ratio of
to
?
Solution
When you see that there are lengths a and 2a, one could think of 30-60-90 triangles. Since all of the other's lengths are a, you could think that .
Drawing the points out, it is possible to have a diagram where
. It turns out that
and
could be the lengths of a 30-60-90 triangle, and the other 3
can be the lengths of an equilateral triangle formed from connecting the dots.
So,
, so