, , , and are points on a line, with the midpoint of segment . The lengths of segments , , and are , , and , respectively. What is the length of segment ?
This is a reading comprehension question as much as it is a mathematical question.
Notice that the assumption that the points are in order from A to B to C to D is challenged by the statement that D is actually exactly in the middle between B and C.
Now, what we want to do is create a diagram that will help us solve the important question – what is the distance between A and D?
Lets start with what we know to be true – D is in the middle between B and C.
Draw a line and two points on it and name them B and C (for the sake of simplicity let’s call the left one B and the right one C). Mark the middle of the segment as point D.
Now point A can be either on the left side of B, between B and D, between D and C, or on the right side of C.
Now, let us look on the distances between the points: BC is 12 – so BD=DC=6 We know AB is 10 – which places A either on the left of B or between D and C (10 is not far enough from B to be on the other side of C). We also know that the distance between A and C is 2 units – this leaves us only one option – A is between D and C – and is located 4 units from D.