AMC10 2014 B

AMC10 2014 B · Q13

AMC10 2014 B · Q13. It mainly tests Triangles (properties), Area & perimeter.

Six regular hexagons surround a regular hexagon of side length 1 as shown. What is the area of $\triangle ABC$?

如图所示，六个正六边形围绕着一个边长为 1 的正六边形。\triangle ABC 的面积是多少？

(A) $2\sqrt{3}$ $2\sqrt{3}$

(B) $3\sqrt{3}$ $3\sqrt{3}$

(D) $2 + 2\sqrt{3}$ $2 + 2\sqrt{3}$

(E) $3 + 2\sqrt{3}$ $3 + 2\sqrt{3}$

Answer

Correct choice: (B)

正确答案：（B）

Solution

Label points E and F as shown in the figure, and let D be the midpoint of BE. Because $\triangle BFD$ is a 30–60–90 triangle with hypotenuse 1, the length of BD is $\sqrt{3}/2$, and therefore BC = $\sqrt{3}$. It follows that the area of $\triangle ABC$ is $\frac{\sqrt{3}}{4} \cdot (\sqrt{3})^2 = 3\sqrt{3}$.

如图标记点 E 和 F，让 D 是 BE 的中点。因为 \triangle BFD 是边长为 1 的 30–60–90 三角形，BD 的长度是 \sqrt{3}/2，因此 BC = \sqrt{3}。于是 \triangle ABC 的面积是 \frac{\sqrt{3}}{4} \cdot (\sqrt{3})^2 = 3\sqrt{3}。

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