AMC12 2001 A

AMC12 2001 A · Q20

AMC12 2001 A · Q20. It mainly tests Coordinate geometry, Transformations.

Points $A = (3,9)$, $B = (1,1)$, $C = (5,3)$, and $D=(a,b)$ lie in the first quadrant and are the vertices of quadrilateral $ABCD$. The quadrilateral formed by joining the midpoints of $\overline{AB}$, $\overline{BC}$, $\overline{CD}$, and $\overline{DA}$ is a square. What is the sum of the coordinates of point $D$?

点 $A = (3,9)$，$B = (1,1)$，$C = (5,3)$，以及 $D=(a,b)$ 位于第一象限，并且是四边形 $ABCD$ 的顶点。连接 $\overline{AB}$、$\overline{BC}$、$\overline{CD}$ 和 $\overline{DA}$ 的中点所形成的四边形是一个正方形。点 $D$ 的坐标之和是多少？

(A) 7 7

(B) 9 9

(D) 12 12

(E) 16 16

Answer

Correct choice: (C)

正确答案：（C）

Solution

We already know two vertices of the square: $(A+B)/2 = (2,5)$ and $(B+C)/2 = (3,2)$. There are only two possibilities for the other vertices of the square: either they are $(6,3)$ and $(5,6)$, or they are $(0,1)$ and $(-1,4)$. The second case would give us $D$ outside the first quadrant, hence the first case is the correct one. As $(6,3)$ is the midpoint of $CD$, we can compute $D=(7,3)$, and $7+3=\boxed{10}$.

我们已知该正方形的两个顶点：$(A+B)/2 = (2,5)$ 和 $(B+C)/2 = (3,2)$。其余两个顶点只有两种可能：要么是 $(6,3)$ 与 $(5,6)$，要么是 $(0,1)$ 与 $(-1,4)$。第二种情况会使 $D$ 不在第一象限，因此应取第一种。由于 $(6,3)$ 是 $CD$ 的中点，可得 $D=(7,3)$，于是 $7+3=\boxed{10}$。

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