AMC10 2017 B

AMC10 2017 B · Q8

AMC10 2017 B · Q8. It mainly tests Triangles (properties), Coordinate geometry.

Points $A(11, 9)$ and $B(2, -3)$ are vertices of $\triangle ABC$ with $AB = AC$. The altitude from $A$ meets the opposite side at $D(-1, 3)$. What are the coordinates of point $C$?

点 $A(11, 9)$ 和 $B(2, -3)$ 是 $\triangle ABC$ 的顶点，且 $AB = AC$。从 $A$ 垂至对边的高与对边交于 $D(-1, 3)$。点 $C$ 的坐标是什么？

(A) (-8, 9) (-8, 9)

(B) (-4, 8) (-4, 8)

(D) (-2, 3) (-2, 3)

(E) (-1, 0) (-1, 0)

Answer

Correct choice: (C)

正确答案：（C）

Solution

The altitude AD lies on a line of symmetry for the isosceles triangle. Under reflection about this line, B will be sent to C. Because B is obtained from D by adding 3 to the x-coordinate and subtracting 6 from the y-coordinate, C is obtained from D by subtracting 3 from the x-coordinate and adding 6 to the y-coordinate. Thus the third vertex C has coordinates (−1 − 3, 3 + 6) = (−4, 9). \n\nOR \n\nTo find the coordinates of C(x, y), note that D is the midpoint of BC. Therefore (x + 2)/2 = −1 and (y − 3)/2 = 3. Solving these equations gives x = −4 and y = 9, so C = (−4, 9).

高线 $AD$ 是等腰三角形的对称轴。关于这条线反射，$B$ 点映射到 $C$ 点。因为从 $D$ 到 $B$ 是 $x$ 坐标加 3，$y$ 坐标减 6，所以从 $D$ 到 $C$ 是 $x$ 坐标减 3，$y$ 坐标加 6。因此第三顶点 $C$ 的坐标为 $(-1-3,3+6)=(-4,9)$。或者求 $C(x,y)$ 的坐标，注意 $D$ 是 $BC$ 的中点。因此 $\frac{x+2}{2}=-1$，$\frac{y-3}{2}=3$。解得 $x=-4$，$y=9$，故 $C=(-4,9)$。

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