Question

which of the following statements must be true based on the diagram below? select all that apply. (diagram is not to scale.)
answer
$overline{ec}$ is a segment bisector.
$overline{ec}$ is a perpendicular bisector.
$overline{ec}$ is an angle bisector.
$e$ is the vertex of a right angle.
$c$ is the vertex of a right angle.
none of the above.

Explanation:

1. Segment bisector: A segment bisector divides a segment into two equal parts. The marks show $DE=EB$, so $\overline{EC}$ splits $\overline{DB}$ into two equal segments, making it a segment bisector.
2. Perpendicular bisector: This requires being both perpendicular and a segment bisector. $\overline{EC} \perp \overline{DB}$ (right angle mark) and it bisects $\overline{DB}$, so it is a perpendicular bisector.
3. Angle bisector: There are no marks showing $\overline{EC}$ splits $\angle DCB$ into two equal angles, so this is not confirmed.
4. E is right angle vertex: The right angle symbol is at $E$, so $\angle DEC$ and $\angle BEC$ are right angles, with $E$ as the vertex.
5. C is right angle vertex: No right angle mark is at $C$, so this is false.

Answer:

$\overline{EC}$ is a segment bisector.
$\overline{EC}$ is a perpendicular bisector.
$E$ is the vertex of a right angle.