AMC10 2019 B

AMC10 2019 B · Q8

AMC10 2019 B · Q8. It mainly tests Triangles (properties), Area & perimeter.

The figure below shows a square and four equilateral triangles, with each triangle having a side lying on a side of the square, such that each triangle has side length 2 and the third vertices of the triangles meet at the center of the square. The region inside the square but outside the triangles is shaded. What is the area of the shaded region?

下图显示了一个正方形和四个等边三角形，每个三角形有一条边位于正方形的一条边上，每个三角形边长为 2，三角形的第三个顶点在正方形的中心。方形内部三角形外部的区域被涂影。涂影区域的面积是多少？

(A) 4 4

(B) $12 - 4\sqrt{3}$ $12 - 4\sqrt{3}$

(D) $4\sqrt{3}$ $4\sqrt{3}$

(E) $16 - 4\sqrt{3}$ $16 - 4\sqrt{3}$

Answer

Correct choice: (B)

正确答案：（B）

Solution

Answer (B): The height of each equilateral triangle is $\sqrt{3}$, so the side length of the square is $2\sqrt{3}$. The area of the square is then $(2\sqrt{3})^2=12$, and the area of the shaded region is $$ 12-4\cdot\frac{\sqrt{3}}{4}\cdot 2^2=12-4\sqrt{3}. $$

答案（B）：每个等边三角形的高为 $\sqrt{3}$，所以正方形的边长为 $2\sqrt{3}$。正方形的面积为 $(2\sqrt{3})^2=12$，阴影部分的面积为 $$ 12-4\cdot\frac{\sqrt{3}}{4}\cdot 2^2=12-4\sqrt{3}. $$

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