A, B, C, and D are points on a line, with D the midpoint of segment . The lengths of segments AB, AC, and BC are 10, 2, and 12, respectively. What is the length of segment ?
A 2
B 4
C 6
D 10
E 12
Answer is B
Here's Why:
The key to this question lies in not jumping to incorrect conclusions. The question names the points on a line. It gives you a variety of information about the points. The one thing it does not do is tell you the order in which the points fall.
Many students assume that the order of the points is A, then B, then C, then D. As you will see, if you try to locate the points in this order, you will be unable to answer the question.
What Is the Question Asking?
The question asks for the length of line segment AD. In order to find this length, you have to establish the relative positions of the four points on the line.
What Do You Know?
Try to draw the figure. You might be tempted to locate point A first. Unfortunately, you don't have enough information about A, yet, to place it.
You can place B, C, and D because D is the midpoint of BC.
You know the lengths of three of the line segments:
AB = 10
AC = 2
BC = 12
Because you know where BC is, you can label the length of BC.
Build the Figure, Adding What You Know and What You Can Figure Out
Because D is the midpoint of BC, you know that BD and DC are each 6 units long.
Where can you place point A?
It has to be 2 units from C, because AC = 2.
It also has to be 10 units from B, because AB = 10.
So the only location for A is between B and C, but closer to C.
Place point A and mark the distances.
It is now an easy matter to figure out the answer to the question:
DC is 6 units.
A is 2 units closer to D than C, so AD is 4 units.
Difficulty: Medium
Question Type: Standard Multiple Choice
(Mathematics)
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