Question

$\overleftrightarrow{pq}$ is constructed by making arcs centered at a and b without changing the compass width. which equation is not necessarily true?
a. $pq = ab$
b. $ap = pb$
c. $aq = bq$
d. $ar = rb$

Explanation:

The construction shown creates the perpendicular bisector of segment $AB$, so $R$ is the midpoint of $AB$ (making $AR=RB$ true). Points $P$ and $Q$ lie on the perpendicular bisector of $AB$, so any point on this line is equidistant from $A$ and $B$, meaning $AP=PB$ and $AQ=BQ$ are true. There is no guarantee that the length of the perpendicular bisector segment $PQ$ equals the length of $AB$.

Answer:

A. $PQ = AB$