Question

1. complete the proof.

given: ( ac = 2(ab) )
prove: ( b ) is the midpoint of ( overline{ac} ).

statements	reasons
2.	2. segment addition postulate
3.	3. transitive property
4.	4. subtraction property of equality
5.	5. definition of congruent segments
6.	6. definition of midpoint

Explanation:

Step1: State the given

We know from the problem that \( AC = 2(AB) \), so this is our first statement.
Statement 1: \( AC = 2AB \)

Step2: Apply Segment Addition Postulate

The Segment Addition Postulate states that for points \( A \), \( B \), \( C \) collinear (which they are as shown in the diagram), \( AC = AB + BC \).
Statement 2: \( AC = AB + BC \)

Step3: Use Transitive Property

Since \( AC = 2AB \) (from Step 1) and \( AC = AB + BC \) (from Step 2), by the Transitive Property (if \( a = b \) and \( b = c \), then \( a = c \)), we can set \( 2AB = AB + BC \).
Statement 3: \( 2AB = AB + BC \)

Step4: Apply Subtraction Property of Equality

Subtract \( AB \) from both sides of the equation \( 2AB = AB + BC \). Using the Subtraction Property of Equality (if \( a = b + c \), then \( a - c = b \)), we get \( 2AB - AB=AB + BC - AB \), which simplifies to \( AB = BC \).
Statement 4: \( AB = BC \)

Step5: Use Definition of congruent segments

If \( AB = BC \), then by the definition of congruent segments (segments with equal length are congruent), \( \overline{AB}\cong\overline{BC} \).
Statement 5: \( \overline{AB}\cong\overline{BC} \)

Step6: Use Definition of midpoint

The definition of a midpoint of a segment \( AC \) is a point \( B \) such that \( \overline{AB}\cong\overline{BC} \) (or \( AB = BC \)). Since we have shown \( \overline{AB}\cong\overline{BC} \), \( B \) is the midpoint of \( \overline{AC} \).
Statement 6: \( B \) is the midpoint of \( \overline{AC} \)

Answer:

1. \( AC = 2AB \)
2. \( AC = AB + BC \)
3. \( 2AB = AB + BC \)
4. \( AB = BC \)
5. \( \overline{AB}\cong\overline{BC} \)
6. \( B \) is the midpoint of \( \overline{AC} \)