2017 AMC 12B Problems/Problem 15
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Problem 15
Let be an equilateral triangle. Extend side
beyond
to a point
so that
. Similarly, extend side
beyond
to a point
so that
, and extend side
beyond
to a point
so that
. What is the ratio of the area of
to the area of
?
Solution
Solution by HydroQuantum
Let .
Recall The Law of Cosines. Letting ,
Since both
and
are both equilateral triangles, they must be similar due to
similarity. This means that
.
Therefore, our answer is .
Solution 2 (inspection)
Note that the height and base of are respectively 4 times and 3 times that of
. Therefore the area of
is 12 times that of
.
By symmetry, . Adding the areas of these three triangles and
for the total area of
gives a ratio of
, or
.