Question
Match the reasons to complete the proof. Given: AB=28, m lies on the line AB Prove: m is the midpoint
Answers
GPT-4o mini
To complete the proof that \( m \) is the midpoint of \( AB \), you would typically follow a logical sequence. Here’s how you might match reasons with statements in the proof:
1. **Given**: \( AB = 28 \)
**Reason**: This is a statement of the given information.
2. **Definition of Midpoint**: A point \( m \) is the midpoint of segment \( AB \) if \( AM = MB \) and \( AB = AM + MB \).
**Reason**: Definition of a midpoint.
3. **Assume \( AM + MB = AB \)**: Since \( m \) lies on line \( AB \), we can express \( AM + MB \) where \( AM \) and \( MB \) are the lengths from \( A \) to \( m \) and \( m \) to \( B \).
**Reason**: Segment addition postulate.
4. **Set up the equation using the given length**: If \( m \) is the midpoint, \( AM = MB \). Hence, \( AM + MB = 2(AM) = 28 \).
**Reason**: Substitution based on the assumption.
5. **Solve for \( AM \)**: \( 2(AM) = 28 \) implies \( AM = 14 \) and \( MB = 14 \).
**Reason**: Algebraic manipulation.
6. **Conclude**: Since \( AM = MB \) and both are equal to \( 14 \), by definition, \( m \) is the midpoint of \( AB \).
**Reason**: The conclusion follows from the definition of a midpoint.
By following these steps, you can successfully complete the proof that \( m \) is indeed the midpoint of \( AB \).
1. **Given**: \( AB = 28 \)
**Reason**: This is a statement of the given information.
2. **Definition of Midpoint**: A point \( m \) is the midpoint of segment \( AB \) if \( AM = MB \) and \( AB = AM + MB \).
**Reason**: Definition of a midpoint.
3. **Assume \( AM + MB = AB \)**: Since \( m \) lies on line \( AB \), we can express \( AM + MB \) where \( AM \) and \( MB \) are the lengths from \( A \) to \( m \) and \( m \) to \( B \).
**Reason**: Segment addition postulate.
4. **Set up the equation using the given length**: If \( m \) is the midpoint, \( AM = MB \). Hence, \( AM + MB = 2(AM) = 28 \).
**Reason**: Substitution based on the assumption.
5. **Solve for \( AM \)**: \( 2(AM) = 28 \) implies \( AM = 14 \) and \( MB = 14 \).
**Reason**: Algebraic manipulation.
6. **Conclude**: Since \( AM = MB \) and both are equal to \( 14 \), by definition, \( m \) is the midpoint of \( AB \).
**Reason**: The conclusion follows from the definition of a midpoint.
By following these steps, you can successfully complete the proof that \( m \) is indeed the midpoint of \( AB \).