¡¡¡¡£¨1£©
¡¡¡¡£¨2£© x = 90
¡¡¡¡Recall that a triangle is a right triangle if and only if the square of the
¡¡¡¡longest side is equal to the sum of the squares of the shorter sides £¨Pythag
¡¡¡¡orean Theorem£©¡£ Hence£¬ £¨1£© implies that the triangle is a right triangle. So
¡¡¡¡the area of the triangle is £¨6£©£¨8£©/2. Note£¬ there is no need to calculate t
¡¡¡¡he area¡ª¡ªwe just need to know that the area can be calculated. Hence£¬ the an
¡¡¡¡swer is either A or D.
¡¡¡¡Turning to £¨2£©£¬ we see immediately that we have a right triangle. Hence£¬ aga
¡¡¡¡in the area can be calculated. The answer is D.
¡¡¡¡Example 3£º Is p < q £¿
¡¡¡¡£¨1£© p/3 < q/3
¡¡¡¡£¨2£© -p + x > -q + x
¡¡¡¡Multiplying both sides of p/3 < q/3 by 3 yields p < q.
¡¡¡¡Hence£¬ £¨1£© is sufficient. As to £¨2£©£¬ subtract x from both sides of -p + x >
¡¡¡¡-q + x£¬ which yields -p > -q.
¡¡¡¡Multiplying both sides of this inequality by -1£¬ and recalling that multiply
¡¡¡¡ing both sides of an inequality by a negative number reverses the inequality
¡¡¡¡£¬ yields p < q.
¡¡¡¡Hence£¬ £¨2£© is also sufficient. The answer is D.
¡¡¡¡Example 4£º If x is both the cube of an integer and between 2 and 200£¬ what i
¡¡¡¡s the value of x£¿
¡¡¡¡£¨1£© x is odd.
¡¡¡¡£¨2£© x is the square of an integer.
¡¡¡¡Since x is both a cube and between 2 and 200£¬ we are looking at the integers
¡¡¡¡£º
¡¡¡¡which reduce to
¡¡¡¡8£¬ 27£¬ 64£¬ 125
¡¡¡¡Since there are two odd integers in this set£¬ £¨1£© is not sufficient to uniqu
¡¡¡¡ely determine the value of x. This eliminates choices A and D.
¡¡¡¡Next£¬ there is only one perfect square£¬ 64£¬ in the set. Hence£¬ £¨2£© is suffic
¡¡¡¡ient to determine the value of x. The answer is B.
¡¡¡¡Example 5£º Is CAB a code word in language Q£¿
