Question

given that d is the midpoint of $overline{ab}$ and b is the midpoint of $overline{ac}$, which statement must be true?
a d b k c
$ad=\frac{1}{3}ac$
$ac = 4db$
$ab + dc=ac$
$bk = kc$

Explanation:

Step1: Use mid - point property for AB

Since D is the mid - point of $\overline{AB}$, we have $AD = DB=\frac{1}{2}AB$.

Step2: Use mid - point property for AC

Since B is the mid - point of $\overline{AC}$, we have $AB = BC=\frac{1}{2}AC$.

Step3: Express AC in terms of DB

From $AB = 2DB$ and $AB=\frac{1}{2}AC$, we can substitute $AB$ in the second equation. So, $2DB=\frac{1}{2}AC$, then $AC = 4DB$.

Step4: Analyze other options

For $AD=\frac{1}{3}AC$, since $AD=\frac{1}{2}AB$ and $AB = \frac{1}{2}AC$, then $AD=\frac{1}{4}AC$, so this option is wrong.
For $AB + DC=AC$, $DC=DB + BC$, $AB = BC$, $AD = DB$, $AC=AD + DB+BC$. Substituting values, $AB + DC=AB+(DB + BC)=AB+(AD + AB)
eq AC$ (in general), so this option is wrong.
For $BK = KC$, there is no information given about the position of K other than it lies on $\overline{AC}$, so we cannot say this is true.

Answer:

B. $AC = 4DB$