Question

points p and q are plotted on a line. a. find a point r so that a 180° rotation with center r sends p to q and q to p. b. is there more than one point r that works for part a?

Explanation:

Step1: Recall rotation property

A 180 - degree rotation about a point $R$ maps a point $P(x_1,y_1)$ to $Q(x_2,y_2)$ such that $R$ is the mid - point of the line segment $\overline{PQ}$.

Step2: Use mid - point formula

The mid - point formula for two points $P(x_1,y_1)$ and $Q(x_2,y_2)$ is $R(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$. In the context of a line, the mid - point of the line segment joining $P$ and $Q$ is the center of the 180 - degree rotation that maps $P$ to $Q$ and $Q$ to $P$.

Step3: Answer part a

The point $R$ is the mid - point of the line segment $\overline{PQ}$. If $P(x_1,y_1)$ and $Q(x_2,y_2)$, then $R=(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$.

Step4: Answer part b

No, there is only one mid - point for the line segment $\overline{PQ}$. Since a 180 - degree rotation about a point $R$ that maps $P$ to $Q$ and $Q$ to $P$ implies that $R$ is the mid - point of $\overline{PQ}$, and the mid - point of a line segment is unique, there is exactly one point $R$ that works for a 180 - degree rotation that maps $P$ to $Q$ and $Q$ to $P$.

Answer:

a. The point $R$ is the mid - point of the line segment joining $P$ and $Q$. If $P$ has coordinates $(x_1,y_1)$ and $Q$ has coordinates $(x_2,y_2)$, then $R$ has coordinates $(\frac{x_1+x_2}{2},\frac{y_1 + y_2}{2})$.
b. No.