Question

given: c is a point on the perpendicular bisector, l, of ab. prove: ac = bc use the drop - down menus to complete the proof. by the unique line postulate, you can draw only one segment, . using the definition of , reflect bc over l. by the definition of reflection, c is the image of itself and is the image of b. since reflections preserve , ac = bc.

Explanation:

1. First blank: Since we are dealing with points \(A\) and \(C\) (to relate to the reflection and the segments \(AC\) and \(BC\)), by the unique line postulate, the segment is \(\overline{AC}\).
2. Second blank: We are performing a reflection. The operation we are using is the definition of a perpendicular bisector (because \(l\) is the perpendicular bisector of \(\overline{AB}\), and reflecting over \(l\) will map \(B\) to \(A\) as per the properties of a perpendicular bisector).
3. Third blank: When we reflect \(B\) over the perpendicular bisector \(l\) (which is the perpendicular bisector of \(\overline{AB}\)), the image of \(B\) is \(A\) (by the property of a perpendicular bisector: any point on the perpendicular bisector is equidistant from the endpoints of the segment it bisects, and reflection over the perpendicular bisector swaps the endpoints).
4. Fourth blank: Reflections are isometries. An isometry is a transformation that preserves distance (lengths of segments). So, since \(AC\) is the image of \(BC\) under a reflection (an isometry), \(AC = BC\) because reflections preserve distance.

Answer:

1. \(\overline{AC}\)
2. perpendicular bisector
3. \(A\)
4. distance