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 \(P\) is equivalent to a reflection over a line \(m\) (where \(m\perp\ell\)) for points with specific symmetry.

• For a \(180^{\circ}\) rotation about \(P\), if \(Q = P\), then rotating \(Q\) \(180^{\circ}\) about \(P\) leaves \(Q\) unchanged. Reflecting \(Q = P\) over line \(m\) also leaves \(Q\) unchanged.
• Let's assume a coordinate - like system. If \(m\) is the \(y\) - axis and \(\ell\) is the \(x\) - axis (intersecting at \(P=(0,0)\)). A point \(Q=(0,y)\) (on line \(m\)):
• Rotating \(Q=(0,y)\) \(180^{\circ}\) about \(P=(0,0)\) gives \(Q'=(0, - y)\). Reflecting \(Q=(0,y)\) over line \(m\) (the \(y\) - axis) gives \(Q'=(0, - y)\)

Step2: Check the general case for part (b)

Let \(P=(0,0)\), \(m\) be the \(y\) - axis (\(x = 0\)), and \(\ell\) be the \(x\) - axis (\(y = 0\)). Consider a point \(Q=(x,y)\) where \(x
eq0\).

• The formula for a \(180^{\circ}\) rotation about the origin \((0,0)\) is \((x,y)\to(-x, - y)\).
• The formula for a reflection over the \(y\) - axis (\(x = 0\)) is \((x,y)\to(-x,y)\). These two transformations are not the same when \(y

eq0\)

Answer:

a. \(Q = P\) and \(Q\) lies on line \(m\)
b. No