Lines

Geometry: Lines and Angles

Read the following SAT test question and then select the correct answer. 

Work all math problems the same way.  Read the problem carefully, identify the bottom line, and assess your options for solving the question.  Choose the most efficient method to attack the problem, and loop back to make sure that your answer matches the bottom line.

Four distinct lines lie in a plane, and exactly two of them are parallel. Which of the following could be the number of points where at least two of the lines intersect?

I. Three

II. Four

III. Five

Bottom Line: # intersections

Assess your Options: You could just start drawing any combination that you can think of, but try to think of the particular answer choices that you are given.  Examine options I, II, and III independently.

Attack the Problem: Think first about option I.  Go ahead and draw out two horizontal parallel lines.  The other lines cannot be parallel to these lines or to each other because the problem says that there are "exactly" two parallel lines.  How could you create three intersections?  One way is to make those next two lines into an “X” and put the middle of the X on one of the preexisting parallel lines.  If you extend the legs of the X out far enough (remember these are lines, not line segments), they will cross the other parallel line in two places.  You have created an image with 3 intersections while following all of the stipulations.

Now turn your attention to option II.  If you have two parallel lines, those lines will never cross.  You know that the other two lines cannot be parallel because this situation has "exactly two" parallel lines.  If you have two lines that are not parallel, no matter how close their slopes are, eventually they must cross.  That is why representing these lines as an X is a good idea.  Any combination of the lines other than the above combination will result in 1 intersection between the lines that are not parallel and 4 intersections where these lines cross the parallel lines for a total of 5 intersections.  There is no way to get only 4 intersections.

Look at option III.  You already thought about it conceptually while examining option II, but you can prove this possibility by drawing a picture.  Draw two horizontal parallel lines.  You can keep the other two lines as an X, but move the middle of the X off of the parallel lines.  Whether the middle of the X is between the parallel lines, above them, or below them, you will now have 5 intersections while following all of the stipulations. (This also confirms your thinking about option II.)

Loop Back:  You examined each option separately, so you are ready to look down at your answer choices.

(A) I only

(B) III only

(C) I and II only

(D) I and III only

(E) I, II, and III

The correct answer is (D).

On sat.collegeboard.org, 29% of the responses were correct.

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