AMC12 2020 A

AMC12 2020 A · Q2

AMC12 2020 A · Q2. It mainly tests Pythagorean theorem, Area & perimeter.

The acronym AMC is shown in the rectangular grid below with grid lines spaced 1 unit apart. In units, what is the sum of the lengths of the line segments that form the acronym AMC?

下面的矩形网格中显示了缩写 AMC，网格线间距为 1 个单位。形成缩写 AMC 的线段长度总和是多少单位？

(A) 17 17

(B) $15 + 2\sqrt{2}$ $15 + 2\sqrt{2}$

(D) $11 + 6\sqrt{2}$ $11 + 6\sqrt{2}$

(E) 21 21

Answer

Correct choice: (C)

正确答案：（C）

Solution

There are 5 horizontal and 8 vertical segments, each of length 1 unit. There are 4 slanted segments, each corresponding to the hypotenuse of an isosceles right triangle with leg length 1. By the Pythagorean Theorem, the length of each slanted segment is $\sqrt{1^{2} + 1^{2}} = \sqrt{2}$. Therefore the sum of all the lengths is $13 + 4\sqrt{2}$ units.

有 5 条水平线段和 8 条垂直线段，每条长 1 单位。有 4 条斜线段，每条对应腿长为 1 的等腰直角三角形的斜边。根据勾股定理，每条斜线段的长度为 $\sqrt{1^{2} + 1^{2}} = \sqrt{2}$。因此所有线段长度总和为 $13 + 4\sqrt{2}$ 单位。

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