AMC8 2007

AMC8 2007 · Q12

AMC8 2007 · Q12. It mainly tests Triangles (properties), Area & perimeter.

A unit hexagram is composed of a regular hexagon of side length 1 and its 6 equilateral triangular extensions, as shown in the diagram. What is the ratio of the area of the extensions to the area of the original hexagon?

一个单位六芒星由边长为 1 的正六边形及其 6 个等边三角形扩展组成，如图所示。扩展部分的面积与原始六边形面积的比率为多少？

(A) 1:1 1:1

(B) 6:5 6:5

(D) 2:1 2:1

(E) 3:1 3:1

Answer

Correct choice: (A)

正确答案：（A）

Solution

(A) Use diagonals to cut the hexagon into 6 congruent triangles. Because each exterior triangle is also equilateral and shares an edge with an internal triangle, each exterior triangle is congruent to each interior triangle. Therefore, the ratio of the area of the extensions to the area of the hexagon is 1:1.

（A）用对角线把六边形分成 6 个全等的三角形。由于每个外侧三角形也是等边三角形，并且与一个内部三角形共享一条边，因此每个外侧三角形都与每个内部三角形全等。因此，延伸部分的面积与六边形面积之比为 1:1。

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