Question

given that d is the midpoint of \\( \overline { a b } \\) and b is the midpoint of \\( \overline { a c } \\), which statement must be true?

\\( \bigcirc a d = \frac { 1 } { 3 } a c \\)
\\( \bigcirc a c = 4 d b \\)
\\( \bigcirc a b + d c = a c \\)
\\( \bigcirc b k = k c \\)

Explanation:

Step1: Use the mid - point property

Since \(D\) is the mid - point of \(\overline{AB}\), then \(AD = DB=\frac{1}{2}AB\). Since \(B\) is the mid - point of \(\overline{AC}\), then \(AB=BC=\frac{1}{2}AC\).

Step2: Relate \(DB\) and \(AC\)

From \(AB = \frac{1}{2}AC\) and \(DB=\frac{1}{2}AB\), substitute \(AB\) in the second equation. We get \(DB=\frac{1}{2}\times\frac{1}{2}AC=\frac{1}{4}AC\), then \(AC = 4DB\).

Step3: Check 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

eq\frac{1}{3}AC\).

• For \(AB + DC=AC\): \(DC=DB + BK+KC\) (assuming \(K\) is a point on \(BC\)), \(AB=\frac{1}{2}AC\), \(AB+DC

eq AC\) in general.

• For \(BK = KC\): There is no information given that \(K\) is the mid - point of \(BC\).

Answer:

\(AC = 4DB\)