Question

if b is the midpoint between segment ac, which of the following equations is correct?
○ none of the choices are correct.
○ ab = ac
○ ab + ac = b
○ ba + bc = ac

Explanation:

Step1: Recall Midpoint Definition

If \( B \) is the midpoint of \( AC \), then \( AB = BC \) and \( AC = AB + BC \). Also, vector or segment addition: \( \overrightarrow{BA} + \overrightarrow{BC} \) – wait, for segments (lengths), \( BA = AB \), \( BC = AB \) (since midpoint), so \( BA + BC = AB + AB = 2AB \), but \( AC = 2AB \), so \( BA + BC = AC \) (in terms of length). Wait, let's check each option:

• Option 1: \( AB = AC \)? No, \( AC = 2AB \), so wrong.
• Option 2: \( AB + AC = B \)? Segments can't sum to a point, wrong.
• Option 3: \( BA + BC = AC \)? Since \( B \) is midpoint, \( BA = AB \), \( BC = AB \), so \( BA + BC = AB + AB = 2AB \), and \( AC = 2AB \), so this holds (for lengths, or vectorially if considering direction, but as segments, length-wise it's correct). Wait, but let's re-express: \( AC = AB + BC \), and \( BA = AB \) (length), \( BC = AB \), so \( BA + BC = AB + AB = AC \) (since \( AC = 2AB \)). Wait, but maybe the question has a typo, but among the options, let's check again. Wait, the options: "None of the choices are correct" or the others. Wait, let's re-express each:

1. \( AB = AC \): False, \( AC = 2AB \).
2. \( AB + AC = B \): False, segments sum to length, not a point.
3. \( BA + BC = AC \): Let's take lengths. Let \( AB = x \), so \( BC = x \), \( AC = 2x \). Then \( BA = x \), \( BC = x \), so \( BA + BC = x + x = 2x = AC \). So this is correct? Wait, but the question says "which of the following equations is correct?" Wait, the original question: "which of the following equations is correct?" (maybe a typo, but let's check the options. Wait, the options are:

• None of the choices are correct.
• \( AB = AC \)
• \( AB + AC = B \)
• \( BA + BC = AC \)

Wait, let's analyze \( BA + BC = AC \). If \( B \) is midpoint, \( A---B---C \), so \( BA \) is from \( B \) to \( A \) (length \( AB \)), \( BC \) is from \( B \) to \( C \) (length \( BC = AB \)). Then \( BA + BC \) (lengths) is \( AB + BC = AB + AB = AC \). So yes, \( BA + BC = AC \) (as lengths) is correct. Wait, but maybe the question is about vectors? If vectors, \( \overrightarrow{BA} + \overrightarrow{BC} \) would be a vector, but the options seem to be about segment lengths (scalars). So among the options, \( BA + BC = AC \) is correct. Wait, but let's check again.

Wait, the first option is "None of the choices are correct" – no, because \( BA + BC = AC \) is correct. Wait, maybe I made a mistake. Wait, \( AC = AB + BC \), and \( BA = AB \) (since \( BA \) is the same length as \( AB \), just opposite direction). So \( BA + BC = AB + BC = AC \). Yes, that's correct. So the correct option is \( BA + BC = AC \). Wait, but let's check the options again. The options are:

• None of the choices are correct.
• \( AB = AC \)
• \( AB + AC = B \)
• \( BA + BC = AC \)

So the last option, \( BA + BC = AC \), is correct. Wait, but maybe the question has a typo, but based on midpoint definition, \( AB = BC \), \( AC = AB + BC \), so \( BA + BC = AB + BC = AC \) (since \( BA = AB \) in length). So that equation is correct.

Step2: Evaluate Each Option

• \( AB = AC \): False, \( AC = 2AB \).
• \( AB + AC = B \): False, segments sum to length, not a point.
• \( BA + BC = AC \): True (as length, \( BA = AB \), \( BC = AB \), so \( BA + BC = AB + AB = AC \)).
• "None of the choices are correct": False, since \( BA + BC = AC \) is correct.

Answer:

\( BA + BC = AC \) (the option with \( BA + BC = AC \))