Question

given that d is the midpoint of (overline{ab}) and k is the midpoint of (overline{bc}), which statement must be true?
diagram: a---d---b---k---c
options:

• b is the midpoint of (overline{ac})
• d bisects (overline{ak})
• (ak + bk = ac)
• (db = bk)

Explanation:

Step1: Analyze each option

• Option 1: B is the midpoint of \( \overline{AC} \)? From the diagram, \( AB \) and \( BC \) are not equal (since D is midpoint of \( AB \), K is midpoint of \( BC \), and segments \( AD, DB, BK, KC \) are on the line, with \( AD = DB \), \( BK = KC \), but \( AB = 2DB \), \( BC = 2BK \), and \( AB

eq BC \) as per the diagram's segment lengths). So B is not the midpoint of \( AC \).

• Option 2: D bisects \( \overline{AK} \)? \( AK = AB + BK \), \( AD = DB \), but \( AD \) and \( DK \) ( \( DK = DB + BK \)) are not equal (since \( DB \) and \( BK \) are not necessarily equal). So D does not bisect \( AK \).
• Option 3: \( AK + BK = AC \)? \( AC = AK + KC \), and since K is midpoint of \( BC \), \( BK = KC \). So \( AK + BK = AK + KC = AC \). This holds.
• Option 4: \( DB = BK \)? There's no information that \( AB = BC \), so \( DB=\frac{1}{2}AB \), \( BK = \frac{1}{2}BC \), and \( AB

eq BC \) (from diagram), so \( DB
eq BK \).

Answer:

\( AK + BK = AC \) (the third option)