proofs about angles quick check
line segment $ \overline { a c } $ is a perpendicular bisector of segment $ \overline { b d } $, with the two segments meeting at point $ e $. what is true of segment $ \overline { b e } $ ? (1 point)
$ \bigcirc $ it must be the same length as segment $ \overline { b c } $
$ \bigcirc $ it must be the same length as segment $ \overline { d e } $
$ \bigcirc $ it must be the same length as segment $ \overline { a b } $
$ \bigcirc $ it must be the same length as segment $ \overline { e a } $

Question

Accepted Answer

# Answer:
It must be the same length as segment $\overline{DE}$

# Explanation:
## Step1: Recall the definition of a perpendicular bisector
A perpendicular bisector of a segment divides the segment into two equal - length parts.
Since $\overline{AC}$ is the perpendicular bisector of $\overline{BD}$ and they meet at point $E$, by the definition of a bisector, $BE = DE$
## Step2: Analyze other options
- For the option “It must be the same length as segment $\overline{BC}$”: There is no information given (from the property of a perpendicular bisector) that $BE=BC$. $BC$ is a side of a triangle (if we consider $\triangle BEC$), and $BE$ is a segment on the bisected segment $\overline{BD}$.
- For the option “It must be the same length as segment $\overline{AB}$”: There is no geometric relationship (from the property of a perpendicular bisector)  that would make $BE = AB$. $AB$ is a separate segment (assuming $A$, $B$ are non - collinear points with respect to the perpendicular bisector property application here)
- For the option “It must be the same length as segment $\overline{EA}$”: $EA$ is a segment on the perpendicular bisector $\overline{AC}$, and there is no geometric rule (from the property of a perpendicular bisector of $\overline{BD}$)  that would equate $BE$ and $EA$ lengths.

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QUESTION IMAGE

Question

Explanation:

Step1: Recall the definition of a perpendicular bisector

Step2: Analyze other options

Answer: