# 2021 CMC 12A Problems/Problem 6

The following problem is from both the 2021 CMC 12A #6 and 2021 CMC 10A #7, so both problems redirect to this page.

## Problem

How many of the following statements are true for every parallelogram $\mathcal{P}$?

      i. The perpendicular bisectors of the sides of $\mathcal{P}$ all share at least one common point.
ii. The perpendicular bisectors of the sides of $\mathcal{P}$ are all distinct.
iii. If the perpendicular bisectors of the sides of $\mathcal{P}$ all share at least one common point then $\mathcal{P}$ is a square.
iv. If the perpendicular bisectors of the sides of $\mathcal{P}$ are all distinct then these bisectors form a parallelogram.


$\textbf{(A) } 0\qquad\textbf{(B) } 1\qquad\textbf{(C) } 2\qquad\textbf{(D) } 3\qquad\textbf{(E) } 4\qquad$

## Solution

Statement (i) is obviously false; suppose $\mathcal{P}$ is a non-rectangular parallelogram, then the statement does not hold.

Statement (ii) is also false; suppose $\mathcal{P}$ is a rectangle, then the perpendicular bisectors of opposite sides are the same.

Statement (iii) is also false; if we let $\mathcal{P}$ be a non-square rectangle, the bisectors still share at least one common point at the center of the rectangle.

Statement (iv) is true. A parallelogram has two pairs of parallel opposite sides. Thus the perpendicular bisectors of those opposite sides will also be parallel, and thus if they are distinct, they will form a parallelogram.

In conclusion, only one statement is true, namely statement (iv), so the answer is $\boxed{\textbf{(B) } 1}$.