Question

2 select the correct answer. $overleftrightarrow{pq}$ 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:

Step1: Recall properties of perpendicular - bisector construction

Since $\overleftrightarrow{PQ}$ is the perpendicular - bisector of segment $AB$ constructed by making arcs centered at $A$ and $B$ with the same compass width. Points $P$ and $Q$ are equidistant from $A$ and $B$ because of the construction method. So, $AP = PB$, $AQ=BQ$, and $AR = RB$ (as $R$ is the mid - point of $AB$).

Step2: Analyze each option

• Option A: There is no reason for $PQ$ to be equal to $AB$. The length of the perpendicular - bisector $PQ$ and the length of the segment $AB$ have no fixed relationship in this construction.
• Option B: In the construction of the perpendicular - bisector, by the property of the intersection of the arcs centered at $A$ and $B$, $AP = PB$.
• Option C: Similarly, $AQ = BQ$ due to the construction of the perpendicular - bisector.
• Option D: Since $\overleftrightarrow{PQ}$ is the perpendicular - bisector of $AB$, $R$ is the mid - point of $AB$, so $AR = RB$.

Answer:

A. $PQ = AB$