Question

1. choose the correct answer.

what is the locus of points in a plane that are equidistant from endpoints a and b?
the angle bisector of ∠ab
the midpoint of (overline{ab})
the vertex of ∠ab
the perpendicular bisector of (overline{ab})

1. choose the correct answer.

what is the locus of points equidistant from the sides of ∠abc?
the angle bisector of ∠abc
the vertex of ∠abc
(overline{ac})
the perpendicular bisector of (overline{ab}) and (overline{bc})

Explanation:

• For the first question:
• The angle bisector of an angle is related to distances from the sides of the angle, not from endpoints of a segment.
• The mid - point is just a single point, not a locus (a set of points).
• The vertex is a single point.
• By the perpendicular bisector theorem, the set of points equidistant from the endpoints \(A\) and \(B\) of a segment \(\overline{AB}\) is the perpendicular bisector of \(\overline{AB}\).
• For the second question:
• The vertex is a single point.
• The segment \(\overline{AC}\) is not related to the locus of points equidistant from the sides of \(\angle ABC\).
• The perpendicular bisector of \(\overline{AB}\) and \(\overline{BC}\) is related to distances from endpoints of segments, not from the sides of an angle.
• By the angle bisector theorem, the set of points equidistant from the sides of an angle \(\angle ABC\) is the angle bisector of \(\angle ABC\).

Answer:

1. The perpendicular bisector of \(\overline{AB}\)
2. The angle bisector of \(\angle ABC\)