Question

lines l and m are perpendicular. a point q has this property: rotating q 180 degrees using center p has the same effect as reflecting q over line m. a. give two possible locations of q.

Explanation:

Step1: Recall reflection and rotation properties

When a point \(Q\) is reflected over a line \(m\) and rotating \(Q\) 180 - degrees about a point \(P\) gives the same result, the point \(P\) is the intersection of lines \(\ell\) and \(m\) and \(Q\) lies on either \(\ell\) or \(m\).
If \(Q\) lies on the line \(m\), when we reflect \(Q\) over \(m\), \(Q\) maps to itself. And if we rotate \(Q\) 180 - degrees about the intersection point \(P\) of \(\ell\) and \(m\), since \(Q\) is on \(m\) and \(\ell\perp m\), the rotation and reflection can be equivalent. Similarly, if \(Q\) lies on the line \(\ell\), the two transformations can also have the same effect.

Answer:

One possible location of \(Q\) is on line \(m\). Another possible location of \(Q\) is on line \(\ell\).