bisecting a segment
given $\overline{ed} \cong \overline{db}$, which statements about the figure are true? check all that apply.
$\square$ $\overline{eb}$ is bisected by $\overline{df}$.
$\square$ a is the midpoint of $\overline{fc}$.
$\square$ $\overline{fc}$ bisects $\overline{db}$.
$\square$ $\overline{eb}$ is a segment bisector.
$\square$ $fa = \frac{1}{2}fc$.
$\square$ $\overline{da} \cong \overline{ab}$.

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Accepted Answer

# Brief Explanations:
1. For " $\overline{EB}$ is bisected by $\overline{DF}$ ": Since $\overline{ED}\cong\overline{DB}$, $D$ is the midpoint of $\overline{EB}$, and $\overline{DF}$ passes through $D$, so $\overline{EB}$ is bisected by $\overline{DF}$ - True.
2. For "A is the midpoint of $\overline{FC}$ ": The marks on $\overline{FC}$ show $FA = AC$, so $A$ is the midpoint - True.
3. For " $\overline{FC}$ bisects $\overline{DB}$ ": $\overline{FC}$ intersects $\overline{DB}$ at $A$, not related to bisecting $\overline{DB}$ (since $\overline{ED}\cong\overline{DB}$ and $D$ is midpoint of $\overline{EB}$, $\overline{FC}$ doesn't bisect $\overline{DB}$) - False.
4. For " $\overline{EB}$ is a segment bisector ": $\overline{EB}$ has $D$ as midpoint of $\overline{EB}$, but $\overline{EB}$ is not a bisector of another segment here ( $\overline{FC}$ is bisected by $\overline{EB}$? Wait, $\overline{EB}$ intersects $\overline{FC}$ at $A$ where $FA = AC$, so $\overline{EB}$ bisects $\overline{FC}$, so $\overline{EB}$ is a segment bisector - True? Wait, no, the definition: a segment bisector is a line, ray, or segment that divides another segment into two equal parts. $\overline{EB}$ intersects $\overline{FC}$ at $A$ with $FA = AC$, so $\overline{EB}$ bisects $\overline{FC}$, so $\overline{EB}$ is a segment bisector - True? Wait, but earlier we thought $D$ is midpoint of $\overline{EB}$. Wait, the options: " $\overline{EB}$ is a segment bisector " - since it bisects $\overline{FC}$, yes. Wait, maybe I made a mistake earlier. Let's re - check:
   - Given $\overline{ED}\cong\overline{DB}$, so $D$ is midpoint of $\overline{EB}$.
   - The marks on $\overline{FC}$ show $FA = AC$, so $A$ is midpoint of $\overline{FC}$.
   - $\overline{EB}$ passes through $A$ (midpoint of $\overline{FC}$), so $\overline{EB}$ bisects $\overline{FC}$, so $\overline{EB}$ is a segment bisector - True.
5. For " $FA=\frac{1}{2}FC$ ": Since $A$ is midpoint of $\overline{FC}$, $FA = AC$ and $FC=FA + AC = 2FA$, so $FA=\frac{1}{2}FC$ - True.
6. For " $\overline{DA}\cong\overline{AB}$ ": There is no information to suggest $DA = AB$, since $D$ is midpoint of $\overline{EB}$, but $A$ is midpoint of $\overline{FC}$, no relation between $DA$ and $AB$ - False.

Wait, let's correct:

1. " $\overline{EB}$ is bisected by $\overline{DF}$ ": $\overline{EB}$ has $D$ as midpoint (since $\overline{ED}\cong\overline{DB}$), and $\overline{DF}$ connects to $D$, so $\overline{DF}$ passes through $D$ (midpoint of $\overline{EB}$), so $\overline{EB}$ is bisected by $\overline{DF}$ - True.

2. "A is the midpoint of $\overline{FC}$ ": The marks on $\overline{FC}$ (the two ticks) show that $FA = AC$, so $A$ is midpoint - True.

3. " $\overline{FC}$ bisects $\overline{DB}$ ": $\overline{FC}$ intersects $\overline{DB}$ at $A$, $\overline{DB}$ has $D$ as midpoint ( $\overline{ED}\cong\overline{DB}$), so $\overline{FC}$ does not bisect $\overline{DB}$ - False.

4. " $\overline{EB}$ is a segment bisector ": $\overline{EB}$ intersects $\overline{FC}$ at $A$ where $FA = AC$, so $\overline{EB}$ bisects $\overline{FC}$, so $\overline{EB}$ is a segment bisector - True.

5. " $FA=\frac{1}{2}FC$ ": Since $A$ is midpoint, $FC = FA+AC = 2FA$, so $FA=\frac{1}{2}FC$ - True.

6. " $\overline{DA}\cong\overline{AB}$ ": No information to support this, $D$ is midpoint of $\overline{EB}$, $A$ is midpoint of $\overline{FC}$, no relation - False.

Wait, but the initial given is $\overline{ED}\cong\overline{DB}$, so $D$ is midpoint of $\overline{EB}$. The segment $\overline{FC}$ has $A$ as midpoint (from the ticks). So:

- " $\overline{EB}$ is bisected by $\overline{DF}$ ": $\overline{EB}$ has midpoint $D$, and $\overline{DF}$ goes through $D$, so yes, $\overline{EB}$ is bisected by $\overline{DF}$ - True.

- "A is the midpoint of $\overline{FC}$ ": Ticks on $\overline{FC}$ at $A$'s sides, so $FA = AC$, so $A$ is midpoint - True.

- " $\overline{FC}$ bisects $\overline{DB}$ ": $\overline{FC}$ intersects $\overline{DB}$ at $A$, not related to bisecting $\overline{DB}$ ( $\overline{DB}$ has midpoint $D$), so False.

- " $\overline{EB}$ is a segment bisector ": $\overline{EB}$ bisects $\overline{FC}$ (since it passes through $A$, midpoint of $\overline{FC}$), so yes - True.

- " $FA=\frac{1}{2}FC$ ": Since $A$ is midpoint, $FA=\frac{1}{2}FC$ - True.

- " $\overline{DA}\cong\overline{AB}$ ": No, $D$ is midpoint of $\overline{EB}$, but $A$ is a point on $\overline{EB}$, no info that $DA = AB$ - False.

Wait, maybe the correct ones are: " $\overline{EB}$ is bisected by $\overline{DF}$ ", "A is the midpoint of $\overline{FC}$ ", " $\overline{EB}$ is a segment bisector ", " $FA=\frac{1}{2}FC$ ". Wait, but let's check the option " $\overline{FC}$ bisects $\overline{DB}$ " again. $\overline{DB}$ is part of $\overline{EB}$, with $D$ as midpoint. $\overline{FC}$ intersects $\overline{EB}$ at $A$, not related to bisecting $\overline{DB}$. So False. And " $\overline{DA}\cong\overline{AB}$ " is False.

Wait, the first option: " $\overline{EB}$ is bisected by $\overline{DF}$ " - $\overline{EB}$ has midpoint $D$, and $\overline{DF}$ connects to $D$, so $\overline{DF}$ bisects $\overline{EB}$ (since it passes through midpoint $D$), so yes, $\overline{EB}$ is bisected by $\overline{DF}$ - True.

Second: "A is the midpoint of $\overline{FC}$ " - ticks on $\overline{FC}$ at $A$'s sides, so $FA = AC$, so $A$ is midpoint - True.

Third: " $\overline{FC}$ bisects $\overline{DB}$ " - $\overline{DB}$ is from $D$ to $B$, $\overline{FC}$ intersects at $A$, no, $\overline{DB}$'s midpoint is $D$? Wait, $\overline{ED}\cong\overline{DB}$, so $D$ is midpoint of $\overline{EB}$, not $\overline{DB}$. Wait, $\overline{DB}$ is a segment from $D$ to $B$, so its midpoint would be a point halfway between $D$ and $B$. $\overline{FC}$ intersects $\overline{DB}$ at $A$, not the midpoint, so False.

Fourth: " $\overline{EB}$ is a segment bisector " - $\overline{EB}$ bisects $\overline{FC}$ (since it passes through $A$, midpoint of $\overline{FC}$), so yes, it's a segment bisector - True.

Fifth: " $FA=\frac{1}{2}FC$ " - since $A$ is midpoint, $FA=\frac{1}{2}FC$ - True.

Sixth: " $\overline{DA}\cong\overline{AB}$ " - no, no information - False.

So the true statements are: " $\overline{EB}$ is bisected by $\overline{DF}$ ", "A is the midpoint of $\overline{FC}$ ", " $\overline{EB}$ is a segment bisector ", " $FA=\frac{1}{2}FC$ ". Wait, but maybe I made a mistake with " $\overline{EB}$ is a segment bisector ". Let's recall the definition: a segment bisector is a line, ray, or segment that divides another segment into two congruent segments. $\overline{EB}$ (a segment) intersects $\overline{FC}$ at $A$, and $FA\cong AC$, so $\overline{EB}$ bisects $\overline{FC}$, so $\overline{EB}$ is a segment bisector - True.

And also, " $FA=\frac{1}{2}FC$ " is true because $A$ is midpoint.

And "A is the midpoint of $\overline{FC}$ " is true.

And " $\overline{EB}$ is bisected by $\overline{DF}$ " is true because $D$ is midpoint of $\overline{EB}$ and $\overline{DF}$ passes through $D$.

Wait, but let's check the option " $\overline{DA}\cong\overline{AB}$ " - there's no marking or given info, so False.

And " $\overline{FC}$ bisects $\overline{DB}$ " - False.

So the correct options are:

- $\boldsymbol{\overline{EB}}$ is bisected by $\boldsymbol{\overline{DF}}$.

- A is the midpoint of $\boldsymbol{\overline{FC}}$.

- $\boldsymbol{\overline{EB}}$ is a segment bisector.

- $\boldsymbol{FA=\frac{1}{2}FC}$.

Wait, but maybe the fourth option " $\overline{EB}$ is a segment bisector " is correct, and the first, second, fourth, fifth.

Wait, let's check a reliable source: a segment bisector is a line, ray, or segment that divides a segment into two equal parts. So if $\overline{EB}$ divides $\overline{FC}$ into two equal parts ($FA = AC$), then $\overline{EB}$ is a segment bisector. So yes.

So the true statements are:

1. $\overline{EB}$ is bisected by $\overline{DF}$.

2. A is the midpoint of $\overline{FC}$.

3. $\overline{EB}$ is a segment bisector.

4. $FA=\frac{1}{2}FC$.

# Answer:
- $\overline{EB}$ is bisected by $\overline{DF}$.
- A is the midpoint of $\overline{FC}$.
- $\overline{EB}$ is a segment bisector.
- $FA=\frac{1}{2}FC$.

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