Question

1. lines ( ell ) 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 ).

b. do all points in the plane have this property?

Explanation:

Step1: Analyze the properties of rotation and reflection

A \(180^{\circ}\) rotation about a point \(P\) and a reflection over a line \(m\) (where \(m\perp\ell\) and \(P\) is the intersection of \(m\) and \(\ell\)):

• For a point \(Q = P\), rotating \(Q = P\) \(180^{\circ}\) about \(P\) leaves \(P\) unchanged (since \(r_{P,180}(P)=P\)). Reflecting \(P\) over line \(m\) also leaves \(P\) unchanged (because the reflection of a point on a line over that line is the point itself, \(r_m(P)=P\)).
• Let's consider a point \(Q\) on line \(\ell\). Let \(P\) be the intersection of \(m\) and \(\ell\). A \(180^{\circ}\) rotation about \(P\) ( \(r_{P,180}(Q)\)): If \(Q\) is on \(\ell\) and \(P\) is the center of rotation, and \(m\perp\ell\) at \(P\), then reflecting \(Q\) over \(m\) ( \(r_m(Q)\)) has the same effect as \(r_{P,180}(Q)\). Using the coordinate - free property: If we assume \(P=(0,0)\), \(m\) is the \(y\) - axis and \(\ell\) is the \(x\) - axis. Let \(Q=(x,0)\). Then \(r_{P,180}(x,0)=(-x,0)\) and \(r_m(x,0)=(-x,0)\)

Step2: Check if all points have the property

Let's use a coordinate system. Let \(P=(0,0)\), \(m\) be the \(y\) - axis (\(x = 0\)) and \(\ell\) be the \(x\) - axis (\(y = 0\)).
Take a point \(Q=(x,y)\) where \(y
eq0\).
The \(180^{\circ}\) rotation about \(P\) is \(r_{P,180}(x,y)=(-x,-y)\).
The reflection over \(m\) (the \(y\) - axis) is \(r_m(x,y)=(-x,y)\). These two transformations \(r_{P,180}(x,y)\) and \(r_m(x,y)\) are equal if and only if \(y=-y\) (i.e., \(y = 0\))

Answer:

a. Two possible locations of \(Q\) are:

• \(Q = P\) (the intersection point of \(m\) and \(\ell\))
• \(Q\) lies on line \(\ell\)

b. No, not all points in the plane have this property. Only points on line \(\ell\) (the line perpendicular to \(m\) at \(P\)) have the property that rotating \(180^{\circ}\) about \(P\) has the same effect as reflecting over \(m\)