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

2022 AMC 12B Problem 10

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10.

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

Place the hexagon on coordinates with its center at the origin

By symmetry all four sides of

GCHF

are equal; compute one distance with the distance formula

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.