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.
$\square$ $fa = \frac{1}{2}fc$
$\square$ $\overline{da} \cong \overline{ab}$

Question

Accepted Answer

# Brief Explanations:
1. For "$\overline{EB}$ is bisected by $\overline{FC}$": Given $\overline{ED}\cong\overline{DB}$, $D$ is the midpoint of $\overline{EB}$, but $\overline{FC}$ intersects $\overline{EB}$ at $A$, not $D$, so this is false.
2. For "A is the midpoint of $\overline{FC}$": The marks on $\overline{FC}$ (equal segments $FA$ and $AC$) show $A$ divides $\overline{FC}$ into two equal parts, so this is true.
3. For "$\overline{FC}$ bisects $\overline{EB}$": $\overline{EB}$ has $D$ as midpoint ($\overline{ED}\cong\overline{DB}$), but $\overline{FC}$ intersects $\overline{EB}$ at $A$, not $D$, so $\overline{FC}$ does not bisect $\overline{EB}$, false.
4. For "$\overline{EB}$ is a segment bisector": A segment bisector divides another segment. $\overline{EB}$ is intersected by $\overline{FC}$ at $A$, but $\overline{EB}$ itself is not bisecting another segment here (since $D$ is its midpoint, but it's not acting as a bisector for another segment in the figure's context), false.
5. For "$FA = \frac{1}{2}FC$": Since $A$ is the midpoint of $\overline{FC}$ (from the segment marks), $FA=AC$ and $FC = FA + AC=2FA$, so $FA=\frac{1}{2}FC$, true.
6. For "$\overline{DA}\cong\overline{AB}$": There's no indication (marks or given info) that $A$ is the midpoint of $\overline{DB}$, so we can't conclude $\overline{DA}\cong\overline{AB}$, false.

# Answer:
A is the midpoint of $\overline{FC}$, $FA = \frac{1}{2}FC$ (the options with these statements, assuming the checkboxes correspond to: the second option, the fifth option, and also let's re - check the last one: Wait, no, earlier mistake. Wait, given $\overline{ED}\cong\overline{DB}$, so $D$ is midpoint of $EB$, and $A$ is midpoint of $FC$ (from the tick marks on $FC$). Also, $\overline{EB}$ is a segment, and $\overline{FC}$ bisects? No, wait, the first option: $\overline{EB}$ is bisected by $\overline{FC}$? No, $D$ is midpoint. Wait, maybe I made a mistake. Let's re - analyze:

Wait, the figure: $E---D---A---B$ on line $EB$, and $F---A---C$ on line $FC$, with $FA = AC$ (tick marks) and $ED = DB$ (given).

1. "$\overline{EB}$ is bisected by $\overline{FC}$": To bisect $\overline{EB}$, the bisector should pass through its midpoint $D$. But $\overline{FC}$ passes through $A$, not $D$, so false.
2. "A is the midpoint of $\overline{FC}$": Since $FA$ and $AC$ have the same tick mark, $FA = AC$, so $A$ is midpoint, true.
3. "$\overline{FC}$ bisects $\overline{EB}$": Bisecting $\overline{EB}$ means dividing it into two equal parts. $\overline{FC}$ intersects $\overline{EB}$ at $A$, but $ED = DB$ (so $D$ is midpoint), $A$ is not $D$, so $\overline{FC}$ does not bisect $\overline{EB}$, false.
4. "$\overline{EB}$ is a segment bisector": A segment bisector is a segment that bisects another segment. $\overline{EB}$ is not shown to bisect any other segment (it's being bisected at $D$ by itself? No, a segment bisector should bisect another segment. So false.
5. "$FA=\frac{1}{2}FC$": Since $A$ is midpoint of $FC$, $FC = FA + AC$ and $FA = AC$, so $FC = 2FA$, so $FA=\frac{1}{2}FC$, true.
6. "$\overline{DA}\cong\overline{AB}$": There's no information (marks or given) that $A$ is midpoint of $DB$, so we can't say $\overline{DA}\cong\overline{AB}$, false.

Also, wait, the first option: "$\overline{EB}$ is bisected by $\overline{FC}$" – no. Wait, maybe the problem's figure has $A$ on $EB$ and $D$ on $EB$ with $ED = DB$, and $A$ on $FC$ with $FA = AC$. Then, is $\overline{FC}$ a bisector of $\overline{EB}$? No, because it doesn't pass through $D$. But is $\overline{EB}$ a bisector of $\overline{FC}$? Wait, $\overline{EB}$ intersects $\overline{FC}$ at $A$, which is the midpoint of $\overline{FC}$. Oh! I made a mistake earlier. So $\overline{EB}$ passes through the midpoint $A$ of $\overline{FC}$, so $\overline{EB}$ bisects $\overline{FC}$, meaning $\overline{FC}$ is bisected by $\overline{EB}$, not the other way. Wait, the first option is "$\overline{EB}$ is bisected by $\overline{FC}$" – no. The third option is "$\overline{FC}$ bisects $\overline{EB}$" – no. The fourth option: "$\overline{EB}$ is a segment bisector" – since it bisects $\overline{FC}$ (passes through its midpoint $A$), so $\overline{EB}$ is a segment bisector (of $\overline{FC}$). And also, $A$ is midpoint of $\overline{FC}$, $FA=\frac{1}{2}FC$, and $\overline{DA}\cong\overline{AB}$? Wait, no, $ED = DB$, but $DA$ and $AB$: unless $A$ is midpoint of $DB$, but we don't know. Wait, let's start over.

Definitions: A segment bisector is a line, ray, or segment that divides another segment into two equal parts (passes through its midpoint). A midpoint of a segment is a point that divides the segment into two equal parts.

Given $\overline{ED}\cong\overline{DB}$, so $D$ is the midpoint of $\overline{EB}$. The marks on $\overline{FC}$ (two equal segments) mean $A$ is the midpoint of $\overline{FC}$ (so $FA = AC$, hence $FA=\frac{1}{2}FC$).

Now, $\overline{EB}$ intersects $\overline{FC}$ at $A$, which is the midpoint of $\overline{FC}$, so $\overline{EB}$ bisects $\overline{FC}$, meaning $\overline{EB}$ is a segment bisector (of $\overline{FC}$).

Also, is $\overline{FC}$ bisecting $\overline{EB}$? No, because it intersects $\overline{EB}$ at $A$, not at $D$ (the midpoint of $\overline{EB}$).

Is $\overline{EB}$ bisected by $\overline{FC}$? No, because $\overline{FC}$ doesn't pass through $D$ (the midpoint of $\overline{EB}$).

Is $A$ the midpoint of $\overline{FC}$? Yes, because $FA = AC$ (from the marks).

Is $FA=\frac{1}{2}FC$? Yes, because $FC = FA + AC$ and $FA = AC$, so $FC = 2FA$ implies $FA=\frac{1}{2}FC$.

Is $\overline{DA}\cong\overline{AB}$? We know $ED = DB$, but there's no information about $DA$ and $AB$. So we can't assume that.

Wait, also, the first option: "$\overline{EB}$ is bisected by $\overline{FC}$" – no. The fourth option: "$\overline{EB}$ is a segment bisector" – yes, because it bisects $\overline{FC}$ (passes through its midpoint $A$).

So let's re - evaluate:

1. $\overline{EB}$ is bisected by $\overline{FC}$: False (bisector of $\overline{EB}$ should pass through $D$, not $A$).
2. A is the midpoint of $\overline{FC}$: True (marks on $\overline{FC}$ show $FA = AC$).
3. $\overline{FC}$ bisects $\overline{EB}$: False (doesn't pass through $D$, midpoint of $\overline{EB}$).
4. $\overline{EB}$ is a segment bisector: True (passes through $A$, midpoint of $\overline{FC}$, so it bisects $\overline{FC}$).
5. $FA=\frac{1}{2}FC$: True (since $A$ is midpoint, $FC = 2FA$).
6. $\overline{DA}\cong\overline{AB}$: False (no info to support).

Wait, now I see my earlier mistake. A segment bisector just needs to pass through the midpoint of another segment. So $\overline{EB}$ passes through $A$ (midpoint of $\overline{FC}$), so $\overline{EB}$ is a segment bisector (of $\overline{FC}$). And $\overline{FC}$ passes through $A$, but $A$ is not the midpoint of $\overline{EB}$ (since $D$ is the midpoint of $\overline{EB}$), so $\overline{FC}$ is not a bisector of $\overline{EB}$, but $\overline{EB}$ is a bisector of $\overline{FC}$.

So the true statements are:

- A is the midpoint of $\overline{FC}$
- $\overline{EB}$ is a segment bisector
- $FA=\frac{1}{2}FC$

Wait, but let's check the original options again:

1. $\overline{EB}$ is bisected by $\overline{FC}$: No (bisected at $D$, not $A$)
2. A is the midpoint of $\overline{FC}$: Yes
3. $\overline{FC}$ bisects $\overline{EB}$: No (bisects at $D$, not $A$)
4. $\overline{EB}$ is a segment bisector: Yes (bisects $\overline{FC}$)
5. $FA=\frac{1}{2}FC$: Yes
6. $\overline{DA}\cong\overline{AB}$: No

So the correct options are the second, fourth, and fifth? Wait, no, the fourth option: "$\overline{EB}$ is a segment bisector" – a segment bisector is a segment that bisects another segment. Since $\overline{EB}$ passes through the midpoint $A$ of $\overline{FC}$, it bisects $\overline{FC}$, so $\overline{EB}$ is a segment bisector.

And the second option: "A is the midpoint of $\overline{FC}$" – yes, from the marks.

Fifth option: "$FA=\frac{1}{2}FC$" – yes, since $A$ is midpoint.

Also, wait, the first option: "$\overline{EB}$ is bisected by $\overline{FC}$" – no, because $\overline{FC}$ doesn't pass through $D$ (the midpoint of $\overline{EB}$).

So the true statements are:

- A is the midpoint of $\overline{FC}$
- $\overline{EB}$ is a segment bisector
- $FA=\frac{1}{2}FC$

But let's check the original problem's options again (the checkboxes):

1. $\overline{EB}$ is bisected by $\overline{FC}$ – no
2. A is the midpoint of $\overline{FC}$ – yes
3. $\overline{FC}$ bisects $\overline{EB}$ – no
4. $\overline{EB}$ is a segment bisector – yes
5. $FA=\frac{1}{2}FC$ – yes
6. $\overline{DA}\cong\overline{AB}$ – no

So the answer is the options:

A is the midpoint of $\overline{FC}$, $\overline{EB}$ is a segment bisector, $FA=\frac{1}{2}FC$ (corresponding to the second, fourth, and fifth checkboxes, assuming the numbering is: 1st: $\overline{EB}$ is bisected by $\overline{FC}$; 2nd: A is the midpoint of $\overline{FC}$; 3rd: $\overline{FC}$ bisects $\overline{EB}$; 4th: $\overline{EB}$ is a segment bisector; 5th: $FA=\frac{1}{2}FC$; 6th: $\overline{DA}\cong\overline{AB}$).

But maybe I was wrong about the fourth option. Let's recall: A segment bisector can be a segment, ray, or line that intersects another segment at its midpoint. So $\overline{EB}$ is a segment, and it intersects $\overline{FC}$ at $A$, which is the midpoint of $\overline{FC}$, so $\overline{EB}$ is a segment bisector (of $\overline{FC}$). So that's true.

And the second option: A is the midpoint of $\overline{FC}$ – true, from the congruent marks on $\overline{FC}$.

Fifth option: $FA=\frac{1}{2}FC$ – true, since $A$ is midpoint, $FC = FA + AC$ and $FA = AC$, so $FC = 2FA$ implies $FA=\frac{1}{2}FC$.

So the correct answers are the second, fourth, and fifth options (assuming the checkboxes are labeled as such). If we list them:

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

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

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

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