2017 AMC 10B Problems/Problem 19
Problem
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 1
Note that by symmetry, is also equilateral. Therefore, we only need to find one of the sides of
to determine the area ratio. WLOG, let
. Therefore,
and
. Also,
, so by the Law of Cosines,
. Therefore, the answer is
Solution 2
As mentioned in the first solution, is equilateral. WLOG, let
. Let
be on the line passing through
such that
is perpendicular to
. Note that
is a 30-60-90 with right angle at
. Since
,
and
. So we know that
. Note that
is a right triangle with right angle at
. So by the Pythagorean theorem, we find
Therefore, the answer is
.
See Also
2017 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 18 |
Followed by Problem 20 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
2017 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 14 |
Followed by Problem 16 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
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