2003 AMC 12A Problems/Problem 14
Problem
Points and lie in the plane of the square such that , , , and are equilateral triangles. If has an area of 16, find the area of .
Solution
Solution 1
Since the area of square is 16, the side length must be 4. Thus, the side length of triangle AKB is 4, and the height of , and thus , is .
The diagonal of the square will then be . From here there are 2 ways to proceed:
First: Since the diagonal is , the side length is , and the area is thus .
Second: Since a square is a rhombus, the area of the square is , where and are the diagonals of the rhombus. Since the diagonal is , the area is .
Solution 2
Because has area , its side length is simply .
Angle chasing, we find that the angle of .
We also know that .
Using Law of Cosines, we find that side .
However, the area of is simply , hence the answer is
Solution 3
First we show that is a square. How? Show that it is a rhombus that has a right angle.
. All the sides of the equilateral triangles are equal, so the triangles are congruent. Notice that , etc, so . So we have a square.
Instead of going to find using Law of Cosines, we can inscribe the square in another bigger one and then subtract the four right triangles in the corners, so after all of this, we find that the answer is
~hastapasta (edited)
See Also
2003 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 13 |
Followed by Problem 15 |
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All AMC 12 Problems and Solutions |
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