AMC12 2020 A

AMC12 2020 A · Q6

AMC12 2020 A · Q6. It mainly tests Transformations, Symmetry.

In the plane figure shown below, 3 of the unit squares have been shaded. What is the least number of additional unit squares that must be shaded so that the resulting figure has two lines of symmetry?

在下面的平面图形中，有 3 个单位正方形被涂黑。必须额外涂黑的最少单位正方形数量是多少，使得最终图形具有两条对称轴？

(A) 4 4

(B) 5 5

(D) 7 7

(E) 8 8

Answer

Correct choice: (D)

正确答案：（D）

Solution

A non-square rectangle has exactly two lines of symmetry in the plane, as shown below. The additional unit squares that must be shaded are indicated with an X in the figure. Indeed, because the square in the second column of the first (topmost) row is shaded, for the figure to have symmetry around the vertical line, the square in the fourth column of the first row must be shaded. Then for the figure to have symmetry around the horizontal line, the squares in the second and fourth columns of the fourth (bottommost) row must be shaded. Similarly, because the square in the lower right corner is shaded, so must be the squares in the other three corners; and because the square in the middle of the third row is shaded, so must be the middle square in the second row. The resulting figure will then have symmetry around both lines. Thus the least number of additional unit squares that must be shaded to give the figure two lines of symmetry is 7.

非正方形矩形在平面中恰有两条对称轴，如下图所示。必须额外涂黑的单位正方形在图中用 X 表示。的确，因为第一行（最顶行）第二列的正方形被涂黑，为了图形围绕垂直线对称，第一行第四列的正方形必须被涂黑。然后为了图形围绕水平线对称，第四行（最底层）第二列和第四列的正方形必须被涂黑。类似地，因为右下角的正方形被涂黑，其他三个角的正方形也必须被涂黑；因为第三行中间的正方形被涂黑，第二行中间的正方形也必须被涂黑。这样得到的图形将同时围绕两条线对称。因此，为了使图形具有两条对称轴，必须额外涂黑的最少单位正方形数量是 7。

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