1976 AMC 12 第 3 题

Question

边长为二的正方形中，从一个顶点到四条边的中点的距离之和为

The sum of the distances from one vertex of a square with sides of length two to the midpoints of each of the sides of the square is

2\sqrt5

2+\sqrt3

2+2\sqrt3

2+\sqrt5

2+2\sqrt5

把所选顶点置于

(0,0)

，其余顶点置于

(2,0)

、

(2,2)

、

(0,2)

Place the chosen vertex at

(0,0)

and the other vertices at

(2,0),

(2,2),

and

(0,2)

两个中点与该顶点相距一个单位，另两个中点分别对应直角边长

2

和

1

Two midpoints are one unit away, while each opposite-side midpoint forms legs

2

and

1

与所选顶点相邻的两条边的中点到该顶点的距离均为

1

。另外两个中点的横、纵坐标差分别为

2

和

1

，所以每个距离都是

\sqrt{2^2+1^2}=\sqrt5

。总和为

2+2\sqrt5

。

所以正确答案是 E。

The two midpoints on sides meeting the chosen vertex are each at distance

1.

The other two midpoints have coordinate differences

2

and

1,

so each is at distance

\sqrt{2^2+1^2}=\sqrt5.

The sum is

2+2\sqrt5.

Therefore, the correct answer is E.

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