2020 AMC 12A Problems/Problem 24
Problem 24
Suppose that is an equilateral triangle of side length
, with the property that there is a unique point
inside the triangle such that
,
, and
. What is
?
Solution
We begin by rotating by
about
, such that in
,
. We see that
is equilateral with side length
, meaning that
. We also see that
is a
right triangle, meaning that
. Thus, by adding the two together, we see that
. We can now use the law of cosines as following:
giving us that . ~ciceronii
Solution 2 (Intuition)
Suppose that triangle had three segments of length
, emanating from each of its vertices, making equal angles with each of its sides, and going into its interior. Suppose each of these segments intersected the segment clockwise to it precisely at its other endpoint and inside
(as pictured in the diagram above). Clearly
and the triangle defined by these intersection points will be equilateral (pictured by the blue segments).
Take this equilateral triangle to have side length . The portions of each segment outside this triangle (in red) have length
. Take
to be the intersection of the segments emanating from
and
. By Law of Cosines,
So,
actually satisfies the conditions of the problem, and we can obtain again by Law of Cosines
- hnkevin42
Video Solution
https://www.youtube.com/watch?v=mUW4zcrRL54
See Also
2020 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 23 |
Followed by Problem 25 |
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