Question

1. lines ( ell ) and ( m ) in the following diagram are perpendicular. a point ( q ) has this property: rotating ( q ) ( 180^{circ} ) using center ( p ) has the same effect as reflecting ( q ) over line ( m ). ( mperpell ) a. give two possible locations of ( q ). b. do all points in the plane have this property? 6. there is a sequence of rigid transformations that takes ( a ) to ( a ), ( b ) to ( b ), and ( c ) to ( c ). the same sequence takes ( d ) to ( d ). draw and label ( d ) in this diagram.

Explanation:

Step1: Recall rotation and reflection properties

For a rotation of 180 - degree about a point \(P\) and reflection over a line \(m\) (where \(m\perp l\) and \(P\) is on \(m\)), if a point \(Q\) is on line \(l\), rotating \(Q\) 180 - degrees about \(P\) is equivalent to reflecting \(Q\) over \(m\). Also, the point \(P\) itself has this property.

Step2: Answer part a

Two possible locations of \(Q\) are: point \(P\) and any point on line \(l\).

Step3: Answer part b

No, not all points in the plane have this property. Only points on line \(l\) and point \(P\) have the property that rotating 180 - degrees about \(P\) is the same as reflecting over \(m\). For a general point \(X\) in the plane, the 180 - degree rotation about \(P\) and reflection over \(m\) will result in different images.

Answer:

a. Point \(P\) and any point on line \(l\)
b. No