Question

The locus of the midpoint of a line segment that is drawn from a given external point $P$ to a given circle with center $O$ and radius $r,$ is:

a straight line perpendicular to $\overline{PO}$

a straight line parallel to $\overline{PO}$

a circle with center $P$ and radius $r$

a circle with center at the midpoint of $\overline{PO}$ and radius $2r$

a circle with center at the midpoint of $\overline{PO}$ and radius $\dfrac12r$

Answer: E

Concepts:transformation midpoint circle

Difficulty rating: 1630

Small Hint:

Let the variable endpoint on the given circle be $X$ and its midpoint with $P$ be $M$

Big Hint:

The map $X\mapsto M$ is a dilation centered at $P$ with scale factor $\frac{1}{2}$

Solution:

As the endpoint $X$ moves on the circle centered at $O,$ its midpoint $M$ with the fixed point $P$ is the image of $X$ under a dilation of scale $\frac{1}{2}$ centered at $P.$ Therefore the locus is a circle whose center is the midpoint of $\overline{PO}$ and whose radius is $\frac{r}{2}.$

Thus, the correct answer is E.

1954 AMC 12 Problem 39

Answer: E

Difficulty rating: 1630

Small Hint:

Big Hint:

Solution:

Problem 39 in Other Years