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: Express \(AC\) in terms of \(DB\)

From \(AB = 2DB\) (because \(AD = DB\)) and \(AC=2AB\) (because \(B\) is the mid - point of \(AC\)).
Substitute \(AB = 2DB\) into \(AC = 2AB\), we get \(AC=2\times(2DB)\).
So \(AC = 4DB\).

Answer:

\(AC = 4DB\)