AMC10 2021 B

AMC10 2021 B · Q9

AMC10 2021 B · Q9. It mainly tests Graphs (coordinate plane), Transformations.

The point $P(a,b)$ in the $xy$-plane is first rotated counterclockwise by $90^\circ$ around the point $(1,5)$ and then reflected about the line $y = -x$. The image of $P$ after these two transformations is at $(-6,3)$. What is $b - a ?$

平面上的点$P(a,b)$先绕点$(1,5)$逆时针旋转$90^\circ$，然后关于直线$y = -x$反射。$P$经过这两个变换后的像位于$(-6,3)$。$b - a$的值是多少？

(A) 1 1

(B) 3 3

(D) 7 7

(E) 9 9

Answer

Correct choice: (D)

正确答案：（D）

Solution

The final image of $P$ is $(-6,3)$. We know the reflection rule for reflecting over $y=-x$ is $(x,y) \rightarrow (-y, -x)$. So before the reflection and after rotation the point is $(-3,6)$. By definition of rotation, the slope between $(-3,6)$ and $(1,5)$ must be perpendicular to the slope between $(a,b)$ and $(1,5)$. The first slope is $\frac{5-6}{1-(-3)} = \frac{-1}{4}$. This means the slope of $P$ and $(1,5)$ is $4$. Rotations also preserve distance to the center of rotation, and since we only "travelled" up and down by the slope once to get from $(-3,6)$ to $(1,5)$ it follows we shall only use the slope once to travel from $(1,5)$ to $P$. Therefore point $P$ is located at $(1+1, 5+4) = (2,9)$. The answer is $9-2 = 7 = \boxed{\textbf{(D)} ~7}$.

$P$的最终像是$(-6,3)$。关于$y=-x$反射的规则是$(x,y) \rightarrow (-y, -x)$。因此，反射前旋转后的点是$(-3,6)$。根据旋转定义，$(-3,6)$与$(1,5)$之间的斜率必须与$(a,b)$与$(1,5)$之间的斜率垂直。第一个斜率是$\frac{5-6}{1-(-3)} = \frac{-1}{4}$。因此，$P$与$(1,5)$的斜率是$4$。旋转还保持到旋转中心的距离，且我们仅沿斜率上下“移动”一次从$(-3,6)$到$(1,5)$，因此从$(1,5)$到$P$也仅使用一次斜率。因此点$P$位于$(1+1, 5+4) = (2,9)$。答案是$9-2 = 7 = \boxed{\textbf{(D)} ~7}$。

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