Question

Regular hexagon $ABCDEF$ has side length $2.$ Let $G$ be the midpoint of $\overline{AB},$ and let $H$ be the midpoint of $\overline{DE}.$ What is the perimeter of $GCHF?$

$4\sqrt3$

$8$

$4\sqrt5$

$4\sqrt7$

$12$

Answer: D

Concepts:regular polygon distance formula coordinate geometry

Difficulty rating: 1500

Solution:

Place the hexagon with center at the origin: $A = (-1, \sqrt3),$ $B = (1, \sqrt3),$ $C = (2, 0),$ $D = (1, -\sqrt3),$ $E = (-1, -\sqrt3),$ $F = (-2, 0).$

Then $G = (0, \sqrt3)$ and $H = (0, -\sqrt3).$ By symmetry all four sides of $GCHF$ are equal, and $GC = \sqrt{2^2 + (\sqrt3)^2} = \sqrt7.$

The perimeter is $4\sqrt7.$

Thus, the correct answer is D.

2022 AMC 12B Problem 10

Answer: D

Difficulty rating: 1500

Solution:

Problem 10 in Other Years