Geometry – Lines and segments

Q: Points D, E, and F are collinear with DE < EF and DF < EF.  If DE = 2x+6, DF = 3x-4, and EF = 22, what is DE?

 Answer: 14


 First, recall that ‘DE’ refers to the length of the segment with endpoints D and E.  It is the distance between D and E.  Next, DON’T ASSUME that E is between the D and F!  Since DE and DF are each less than EF, and all three points lie on the same line (collinear), D must be between E and F.

 Draw the diagram:


E                         D                   F

 Now label the distances as they are given in the problem:



E        2x+6         D     3x-4      F

 Use the concept: Part + Part = Whole (also called the Segment Addition Postulate) to write the equation:

 2x+6 + 3x-4 = 22


5x + 2 = 22

5x = 20

x = 4

 Plug x=4 back in to the ‘DE’ part:

DE = 2x+6 = 2(4) + 6 = 14.

 Tip:  To be sure your answer is correct, plug x=4 into the ‘DF’ part, too:

DF = 3x-4 = 3(4)-4 = 8.  Now, see that the sum of the two ‘parts’ does equal the ‘whole’:

14 + 8 = 22.

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