Question

(primary, unit 1, lesson 11) 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. m ⊥ l a. give two possible locations of q. type your response in the space below. b. do all points in the plane have this property? type your response in the space below.

Explanation:

Step1: Understand rotation and reflection properties

For a point \(Q\), rotating it 180 - degrees about point \(P\) (the intersection of perpendicular lines \(l\) and \(m\)) is equivalent to reflecting it over line \(m\) in certain cases. If a point \(Q\) lies on line \(l\), rotating it 180 - degrees about \(P\) has the same effect as reflecting it over line \(m\).

Step2: Answer part a

Two possible locations of \(Q\) are any two distinct points on line \(l\). For example, a point \(Q_1\) to the left of \(P\) on line \(l\) and a point \(Q_2\) to the right of \(P\) on line \(l\).

Step3: Answer part b

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

Answer:

a. Any two points on line \(l\).
b. No