Question

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

\\(ad = \frac{1}{3}ac\\)
\\(ac = 4db\\)
\\(ab + dc = ac\\)
\\(bk = kc\\)

Explanation:

Define segment relations using midpoints

Using the Rays and Segments knowledge point

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Express all lengths in terms of a single variable

Let \(AD = x\).

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Evaluate each statement

We test the given options using \(x\):

• Option 1: \(AD = \frac{1}{3}AC \implies x = \frac{1}{3}(4x) \implies 1 = \frac{4}{3}\) (False)
• Option 2: \(AC = 4DB \implies 4x = 4(x) \implies 4x = 4x\) (True)
• Option 3: \(AB + DC = AC \implies 2x + 3x = 4x \implies 5x = 4x\) (False)
• Option 4: \(BK = KC\) (We have no information defining \(K\) as a midpoint of \(\overline{BC}\), so this is not necessarily true)

Answer:

• (A) \(AD = \frac{1}{3}AC\)
• (B) \(AC = 4DB\) (Correct answer)
• (C) \(AB + DC = AC\)
• (D) \(BK = KC\)