Question

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

Explanation:

Step1: Use mid - point property

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

Step2: Express \(AC\) in terms of \(DB\)

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

Let's 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 \(AB + DC=AC\): \(DC=DB + BC\), \(AB + DC=AB+(DB + BC)\). Since \(AB = BC\) and \(AB = 2DB\), \(AB+DC=2DB+DB + 2DB=5DB

eq AC = 4DB\).

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

Answer:

AC = 4DB