AMC 8 1999 Problem 23

AMC8 1999 年第 23 题

四边形与多边形★★★☆☆

Square

ABCD

has sides of length 3. Segments

CM

and

CN

divide the square's area into three equal parts. How long is segment

CM

正方形

ABCD

的边长为 3。线段

CM

和

CN

将正方形的面积分成相等的三部分。线段

CM

的长度是多少？

正确答案 · Correct Answer

解析 · Solution

Solution 1

Since the square has side length 3, the area of the entire square is 9.

The segments divide the square into 3 equal parts, so the area of each part is

9 \div 3 = 3

Since

\triangle CBM

has area 3 and base CB = 3, using the area formula for a triangle:

\displaystyle A_{tri} = \frac{1}{2} bh

\displaystyle 3 = \frac{1}{2} \cdot 3h

, h = 2

Thus, height BM = 2.

Since

\triangle CBM

is a right triangle,

CM = \sqrt{BM^2 + BC^2} = \sqrt{2^2 + 3^2} = \text{(C) } \sqrt{13}

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