Question

given that d is the midpoint of \\(\overline{ab}\\) and k is the midpoint of \\(\overline{bc}\\), which statement must be true?
options: b is the midpoint of \\(\overline{ac}\\); \\(ak + bk = ac\\); \\(db \cong bk\\); d bisects \\(\overline{ak}\\)

Explanation:

Step1: Analyze given midpoints

D is midpoint of \( \overline{AB} \), so \( AD = DB \). K is midpoint of \( \overline{BC} \), so \( BK = KC \).

Step2: Evaluate each option

• Option 1: B is midpoint of \( \overline{AC} \)? \( AB \) and \( BC \) lengths: \( AB = 2AD \), \( BC = 2BK \). Not necessarily \( AB = BC \), so B not midpoint.
• Option 2: \( DB = BK \)? \( DB = \frac{AB}{2} \), \( BK = \frac{BC}{2} \). No info \( AB = BC \), so not true.
• Option 3: \( AK + BK = AC \)? \( AK = AB + BK \), so \( AK + BK = AB + 2BK \). \( AC = AB + BC = AB + 2BK \) (since \( BC = 2BK \)). So \( AK + BK = AC \) holds.
• Option 4: D bisects \( \overline{AK} \)? \( AD = DB \), but \( DB \) and \( BK \) not equal (unless \( AB = BC \)), so \( AD

eq DK \), D doesn't bisect \( AK \).

Answer:

\( \boldsymbol{AK + BK = AC} \) (the option with this statement)