Question

line cd bisects $overline{ab}$ at point g. if $ae = be$, which equation must be true?
a. $be = bg$
b. $ae = 2(bg)$
c. $ae = bg$
d. $ag = bg$
e. none of these

Explanation:

Step1: Recall definition of segment bisector

A segment bisector divides a segment into two equal parts. Since line $CD$ bisects $\overline{AB}$ at $G$, this means $G$ is the midpoint of $\overline{AB}$.

Step2: Apply midpoint property

By the definition of a midpoint, the two resulting segments are congruent:
$AG = BG$

Step3: Verify other options

• Option A: $BE$ is a slant segment, $BG$ is a horizontal segment; they are not equal.
• Option B: $AE$ is a slant segment, $2(BG)$ is the length of $\overline{AB}$; they are not equal.
• Option C: $AE$ is a slant segment, $BG$ is a horizontal segment; they are not equal.

Answer:

D. $AG = BG$