每日一题

Mathematics>Standard Multiple Choice

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

Which of the following statements must be true of the lengths of the segments on line m above?

I. AB+CD=AD
II. AB+BC=AD-CD
III. AC-AB=AD-CD

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

答案和解析

答案：B

解析：

Consider each statement separately. For example, consider statement I, A B+C D=A D. From the figure, you can see that segment A D is made up of the segments A B, B C, and C D. This tells you that A B+C D cannot equal A D, since B C cannot equal zero. Statement I is not true.

Consider statement II, A B+B C=A D-C D. Since B is between A and C, it follows that A B+B C=A C. Since C is between A and D, it follows that A C+C D=A D. Therefore, A D-C D=A C. Since both A B+B C and A D-C D equal A C, they are equal to each other. Statement II is true.

Consider statement III, A C-A B=A D-C D. The left side of the equation, A C-A B, is equivalent to B C. The right side of the equation, A D-C D, is equivalent to A C. Since A B cannot equal zero, B C is not equal to A C. Statement III is not true.

Statement II is the only one that is true.