2003 AMC 12A Problem 14

Problem 14 of 25IntermediateGeometry

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.

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Solution

The square

ABCD

has side

4.

By the

90^\circ

rotational symmetry,

KLMN

is also a square. Each equilateral triangle on a side of length

4

has height

2\sqrt3,

so the diagonal

KM=4+2(2\sqrt3)=4+4\sqrt3.

A square with diagonal

d

has area

\dfrac12 d^2,

[KLMN]=\dfrac12(4+4\sqrt3)^2

=32+16\sqrt3.

Thus, the correct answer is D.