Regular hexagon ABCDEF has vertices A and C at (0, 0) and (7, 1), respectively. What is its area?

2006 AMC 12B Problem 16

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16.

Regular hexagon

ABCDEF

has vertices

A

and

C

(0, 0)

and

(7, 1),

respectively. What is its area?

20\sqrt{3}

22\sqrt{3}

25\sqrt{3}

27\sqrt{3}

50

A

and

C

are two vertices apart, so

AC

is a short diagonal of the hexagon

For side length

s,

the short diagonal is

s\sqrt3;

the area is

\dfrac{3\sqrt3}{2}s^2

The distance is

AC = \sqrt{7^2 + 1^2} = \sqrt{50}.

In a regular hexagon with side

s,

the distance between vertices two apart is

s\sqrt3,

s^2 \cdot 3 = 50,

giving

s^2 = \dfrac{50}{3}.

The hexagon’s area is

\frac{3\sqrt3}{2}s^2 = \frac{3\sqrt3}{2} \cdot \frac{50}{3} = 25\sqrt3.

Thus, the correct answer is C.