AMC8 1999

AMC8 1999 · Q23

AMC8 1999 · Q23. It mainly tests Pythagorean theorem, Area & perimeter.

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 边长为 3。线段 CM 和 CN 将正方形的面积分为三等分。线段 CM 长多少？

(A) $\sqrt{10}$ $\sqrt{10}$

(B) $\sqrt{12}$ $\sqrt{12}$

(D) $\sqrt{14}$ $\sqrt{14}$

(E) $\sqrt{15}$ $\sqrt{15}$

Answer

Correct choice: (C)

正确答案：（C）

Solution

Answer (C): One-third of the square's area is 3, so triangle $MBC$ has area $3=\frac{1}{2}(MB)(BC)$. Since side $BC$ is 3, side $MB$ must be 2. The hypotenuse $CM$ of this right triangle is $\sqrt{2^2+3^2}=\sqrt{13}$.

答案（C）：正方形面积的三分之一是 3，因此三角形 $MBC$ 的面积为 $3=\frac{1}{2}(MB)(BC)$。由于边 $BC$ 为 3，所以边 $MB$ 必须为 2。该直角三角形的斜边 $CM$ 为 $\sqrt{2^2+3^2}=\sqrt{13}$。

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