Question

the perpendicular bisector theorem. consider pc which is the perpendicular bisector of rb. recall that when reflecting a point across a line of reflection, the line of reflection forms the perpendicular bisector between the pre - image and the image. the reflection of point p across pc is point p by the definition of reflection. the reflection of point r across pc is point b. therefore, pr = pb. the perpendicular bisector theorem states: \any point on the perpendicular bisector of a segment is equidistant from the endpoints of that segment.\ study the example and then answer each question. decide whether each statement about the example is true or false. true : rc = cb false : pc = pr false : pc = pr true : m∠rpc = m∠bpc use the diagram on the left to answer each question. if point f lies on line pc, then fr = if point g lies on line pc, then gb = any point on line pc is the same distance from the points consider point q which is the same distance from points a ∠aqb. use the diagram below to answer the remaining ques

Explanation:

Step1: Recall perpendicular - bisector property

The perpendicular bisector of a line segment has the property that any point on it is equidistant from the endpoints of the segment. In the case of $\overline{RB}$ with perpendicular bisector $\overleftrightarrow{PC}$, we know that for any point on $\overleftrightarrow{PC}$, the distances to $R$ and $B$ are equal.

Step2: Analyze angle - equality

Since $\overleftrightarrow{PC}$ is the perpendicular bisector of $\overline{RB}$, $\triangle RPC$ and $\triangle BPC$ are congruent (by SSS - Side - Side - Side congruence criterion as $RC = CB$, $PC=PC$, and $RP = PB$). So, $\angle RPC=\angle BPC$, which means $m\angle RPC = m\angle BPC$ is True.

Step3: Analyze distance - equality

$PC$ is a line on the perpendicular bisector, but $PC$ is not equal to $PR$. $PR$ is the distance from a point $P$ (not on the endpoints of $\overline{RB}$) to an endpoint of the segment $\overline{RB}$, and $PC$ is part of the perpendicular bisector. So, $PC = PR$ is False.

Step4: Analyze segment - equality

As $\overleftrightarrow{PC}$ is the perpendicular bisector of $\overline{RB}$, it divides $\overline{RB}$ into two equal parts. So, $RC = CB$ is True.

Step5: Analyze distance from a point on the perpendicular bisector to endpoints

If a point $F$ lies on line $PC$ (the perpendicular bisector of $\overline{RB}$), then by the definition of a perpendicular bisector, $FR=FB$.

Step6: Analyze distance from a point on the perpendicular bisector to endpoints

If a point $G$ lies on line $PC$ (the perpendicular bisector of $\overline{RB}$), then $GB = GR$.

Step7: Analyze distance from a point to two other points

For a point $Q$ which is the same distance from points $A$ and $B$, we cannot make further conclusions about its relation to the given perpendicular - bisector $\overleftrightarrow{PC}$ without more information about the position of $A$ and $B$ relative to $\overline{RB}$ and $\overleftrightarrow{PC}$.

Answer:

True : $m\angle RPC = m\angle BPC$
False : $PC = PR$
True : $RC = CB$
True : If point $F$ lies on line $PC$, then $FR = FB$
True : If point $G$ lies on line $PC$, then $GB = GR$