Question

if point b bisects the line segment ac, then...
e ( ac = 2ab )
f all of the above
g ( xz = 2yz )
h ( ab = bc )

Explanation:

Step1: Recall the definition of a midpoint

A point that bisects a line segment divides it into two equal parts. So if point \( B \) bisects line segment \( AC \), then \( AB \) and \( BC \) are equal in length, i.e., \( AB = BC \). Also, since \( AC=AB + BC \) and \( AB = BC \), we can substitute \( BC \) with \( AB \) to get \( AC=AB + AB=2AB \). So both \( AB = BC \) and \( AC = 2AB \) are true, which means "All of the above" (option F) is correct? Wait, let's check the options again. Wait, the options are: E: \( AC = 2AB \), F: All of the above, G: \( XZ=2YZ \) (this is about a different segment, so irrelevant), H: \( AB = BC \).

First, by the definition of a midpoint (bisecting a segment), \( AB=BC \) (H is correct). Then, since \( AC=AB + BC \) and \( AB = BC \), substituting \( BC \) with \( AB \) gives \( AC=AB+AB = 2AB \) (E is correct). So both E and H are correct, so "All of the above" (F) would be correct? Wait, but G is about \( XZ \) and \( YZ \), which is a different segment, so maybe there's a typo, but assuming the question is about \( AC \) bisected by \( B \), then:

- H: \( AB = BC \) is true (definition of bisecting a segment: dividing into two equal parts).
- E: \( AC=2AB \) is true because \( AC = AB + BC \) and \( AB = BC \), so \( AC=AB + AB = 2AB \).
- G is about a different segment, so it's not relevant, but since E and H are true, and F says "All of the above", but wait, maybe the original problem had a typo, but based on the given, if we consider that E and H are correct, then F (All of the above) would be correct? Wait, no, G is about \( XZ \) and \( YZ \), which is not related to \( AC \) and \( B \), so maybe that's an error. But assuming that the options are about \( AC \) bisected by \( B \):

So let's re - evaluate:

If \( B \) bisects \( AC \), then:

1. \( AB = BC \) (by definition of a bisector of a segment, it divides the segment into two congruent parts) - so H is correct.

2. \( AC=AB + BC \), and since \( AB = BC \), then \( AC=AB + AB=2AB \) - so E is correct.

Since E and H are correct, and F is "All of the above", and G is about a different segment (so maybe it's a mistake, but if we assume that the options are supposed to be about \( AC \) and \( B \)), then F would be correct? Wait, but maybe the original problem had a mistake in option G. Alternatively, maybe the intended answer is F (All of the above) because E and H are correct, and maybe G is a distractor or a typo.

Wait, let's check the options again:

- E: \( AC = 2AB \)

- F: All of the above

- G: \( XZ = 2YZ \)

- H: \( AB = BC \)

So, for the given problem (point \( B \) bisects \( AC \)):

- H is correct (bisecting means \( AB = BC \))

- E is correct (since \( AC=AB + BC \) and \( AB = BC \), so \( AC = 2AB \))

- G is about a different segment, so it's not correct. But if F is "All of the above", but G is incorrect, then maybe there's a mistake. Wait, maybe the original problem had a different label, like maybe \( Y \) bisects \( XZ \), but in the given problem, it's about \( B \) bisecting \( AC \). So perhaps it's a typo in option G. Assuming that the options are supposed to be related to \( AC \) and \( B \), then E and H are correct, so "All of the above" (F) would be correct if G is also correct, but since G is about a different segment, maybe the intended answer is F (maybe G is a mistake). Alternatively, maybe the user made a typo, but based on the given, let's proceed.

So, the correct options are E, H, and if we consider F as "All of the above" (assuming G is a mistake), then F is correct. But let's check the definitions again.

Bisecting a segment means dividing it into two equal parts, so \( AB = BC \) (H is correct). Then \( AC=AB + BC=AB + AB = 2AB \) (E is correct). So E and H are correct, so "All of the above" (F) is correct (assuming G is a distractor or a typo).

Answer:

F. All of the above, E. \( AC = 2AB \), H. \( AB = BC \) (But the most appropriate single option is F as it includes E and H which are correct for the given problem of \( B \) bisecting \( AC \))