2003 AMC 12A Problem 14

Below is the professionally curated solution for Problem 14 of the 2003 AMC 12A, 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.

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Concepts:equilateral triangle square (geometry)symmetry

Difficulty rating: 1570

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 $KLMN.$

$32$

$16 + 16\sqrt{3}$

$48$

$32 + 16\sqrt{3}$

$64$

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 $\dfrac12 d^2,$ so $[KLMN]=\dfrac12(4+4\sqrt3)^2=32+16\sqrt3.$

Thus, the correct answer is D.

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