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Mathematics>Standard Multiple Choice

Read the following SAT test question and then click on a button to select your answer.

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

(A) I only
(B) III only
(C) I and II only
(D) I and III only
(E) I, II, and III

答案和解析

答案：D

解析：

Since exactly two of the four lines are parallel, the other two lines are not parallel. If the two non-parallel lines intersect at a point that is not on either one of the parallel lines, then the configuration of lines will give a total of 5 points of intersection. (The best way to verify this is by drawing the two parallel lines and then putting in the other two lines.) If, on the other hand, the two non-parallel lines intersect at a point that is on one of the parallel lines, then there will be a total of 3 points of intersection in the figure. (Again, a sketch is the best way to verify this.) Any arrangement of the four lines will again yield either 3 or 5 points of intersection. Since you can’t obtain four points of intersection, the correct answer is I and III only.