AbdurRakib wrote:
A, B, C, and D are points on a line. If C is the midpoint of line segment AB and if D is the midpoint of line segment CB, is the length of line segment DB greater than 5?
1) The length of line segment AC is greater than 8
2) The length of line segment CD is greater than 6
We are given that A, B, C, D are points on a line. We also also given that C is the midpoint of line segment AB and D is the midpoint of line segment CB. We need to determine whether the length of line segment DB is greater than 5. Let’s denote the length of a line segment using the absolute value sign. Therefore, the question becomes: Is |DB| > 5?
Since C is the midpoint of line segment AB, C must be between A and B. Furthermore, since D is the midpoint of line segment CB, D must be between C and B. Therefore, the points lie on the line in the following order: A, C, D, B.
Since D is the midpoint of line segment CB, we have |CD| = |DB|. Notice that |CB| = |CD| + |DB|. Since |CD| = |DB|, that means |CB| = 2|DB|. Moreover, since C is the midpoint of line segment AB, |AC| = |CB|. Since |CB| = 2|DB|, |AC| = 2|DB|.
Statement One Alone:The length of line segment AC is greater than 8.
Since we know |AC| = 2|DB|, we know that 2|DB| > 8 and thus|DB| > 4 However, knowing |DB| is greater than 4 does not mean it is greater than 5. Statement one alone is not sufficient to answer the question. We can eliminate answer choices A and D.
Statement Two Alone:The length of line segment CD is greater than 6.
Recall that |CD| = |DB|. Therefore, if |CD| > 6, then |DB| > 6 and thus |DB| > 5. Statement two alone is sufficient to answer the question.
Answer: B
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