Question

study the example and then answer each question. decide whether each statement about the example is true or false. 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 and b. use the diagram below to answer the remaining questions. consider point q which is the same distance from points a and b. use the diagram below to answer the remaining questions. true false : m∠rpc = m∠bpc true false : pc = pr true false : rc = cb

Explanation:

Step1: Recall properties of reflection

In a reflection, the line of reflection is the perpendicular - bisector of the segment joining a point and its image. Points on the line of reflection are equidistant from corresponding points of the pre - image and image.

Step2: Analyze the angle - equality statement

If PC is the line of reflection, then $\angle RPC$ and $\angle BPC$ are angles formed by the line of reflection and the segments joining points to their images. Since PC is the line of reflection, the angles of incidence and reflection are equal, so $m\angle RPC=m\angle BPC$. This statement is True.

Step3: Analyze the segment - equality statement

Just because a point lies on the line of reflection (PC), it does not mean that $PC = PR$. PC is a segment on the line of reflection and PR is a segment from a point (P) to its image - related point (R). There is no reason for them to be equal. This statement is False.

Step4: Analyze the other segment - equality statement

There is no information or property of reflection that would imply $RC = CB$. Points R, C, and B are not related in such a way that these two segments must be equal. This statement is False.

Step5: Analyze distance - from - points statement

If a point Q is on the perpendicular bisector of AB (the line of reflection), then Q is equidistant from A and B. But the problem does not state that PC is the perpendicular bisector of AB. For a point on the line of reflection PC, any point on line PC is the same distance from the pre - image and image points related by the reflection. If we assume the correct context of reflection, for a point on the line of reflection PC, it is the same distance from corresponding pre - image and image points. But the way the statement is worded about points A, C and R etc. is not correct in general. However, if we consider the property of a point on the line of reflection being equidistant from pre - image and image points, we know that for a point on the line of reflection PC, it is the same distance from the points related by reflection. But the statement about points A, C and R is mis - worded. If we assume the correct concept of reflection, a point on the line of reflection is equidistant from corresponding pre - image and image points.
If point G lies on line PC (the line of reflection), then the distance from G to the pre - image and image points (related by reflection) are equal. But we don't know the relationship between GB and other segments in a way that GB would be equal to some other segment just because G is on PC. The statement "If point G lies on line PC, then GB=" is incomplete. Similarly for "If point F lies on line PC, then FR=". But if we consider the property of reflection, a point on the line of reflection is equidistant from the pre - image and image points.

Answer:

True : $m\angle RPC=m\angle BPC$
False : $PC = PR$
False : $RC = CB$