On a number line, points A, D, B, E, C, and F are arranged from left to right. Point B is the midpoint of segment AC, and point E is the midpoint of segment DF.

Question

If length BE is less than length EC, which of the following statements are true? (multiple choice, pick one only)

DB < EC

BE > CF

BC + DE < AD + EF

AD + AB < EC + EF

none are true

Answer 1

First, some strategy: In problems of this type, definitely, draw the indicated figure! If it has been drawn for you, it may not be drawn so as to be easily manipulated; redraw it in that case (remember that provided figures may be deliberately mis-scaled so as to mislead you). Then, join the related point sets by arcs, say above the line for ABC and below the line for DEF. This makes it quite easy to manipulate the two point sets: DEF has moved a bit left, by comparison with ABC. So the order of business is: (1) redraw the figure to permit recognizing relative distances; (2) reason from distance pairs; (3) discard equal parts to test provided equations.

Now, the tactics: With the figure drawn as suggested, draw arrows to the left from D, E, and F to help you keep in mind that all three move together (since otherwise the midpoint relationship fails!). Start the process with the additional arbitrary condition that AC = DF. Then it's quite easy to see that DB = EC, BE = CF, BC + DE = AD + EF, and AD + AB = EC + EF. Thus, “none are true.” Remember that "are true" means "must be true in all cases"!

Note that if the prospective answers include non-equivalent terms in their inequality, then a question might instead be asked “which statement(s) might be true?”. In this case, first cancel any pairs of equivalent terms from the sides of the statements, so that the remaining terms may be more easily examined; any displacement of one point set vs. the other that could make the residual statements true, but doesn't violate the problem conditions (by reordering the points, for example!), is then an answer. You might start with the additional condition that AC = DF, examine for possibly true statements as above, then relax the relationship to include also the AC > DF and AC < DF possibilities. You must be systematic in examining all the permitted types of inequalities; some instances may require drawing extra figures to check them out since possible geometric relationships may vary considerably from those in your initial sketch.

-- Cheers, -- Mr. d.