Problem:
Points K,L,M, and N lie in the plane of the square ABCD so that AKB, BLC,CMD, and DNA are equilateral triangles. If ABCD has an area of 16 , find the area of KLMN.
Answer Choices:
A. 32
B. 16+163​
C. 48
D. 32+163​
E. 64 Solution:
Quadrilateral KLMN is a square because it has 90∘ rotational symmetry, which implies that each pair of adjacent sides is congruent and perpendicular. Since ABCD has sides of length 4 and K is 23​ from side AB, the length of the diagonal KM is 4+43​. Thus the area is
21​(4+43​)2=32+163​​
\section*{OR}
Note that m(∠NAK)=150∘. By the Law of Cosines,
(NK)2=42+42−2(4)(4)(−23​​)=32+163​
Since KLMN is a square, its area is (NK)2=32+163​​.
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