2003 AMC 12A Problem 14

Attempt Problem 14 of the 2003 AMC 12A below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2003 AMC 12A solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

14.

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.

32

16 + 16\sqrt{3}

48

32 + 16\sqrt{3}

64

KLMN

is a square by the rotational symmetry of the figure

Its diagonal

KM

equals the square’s side plus twice the triangle height

2\sqrt3

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.

Problem 14 in other years