2. 外语
  3. What is the length of the hypotenuse of △ABC? (1) The lengths of the three sides of △ABC are consecutive even integers. (2) The

What is the length of the hypotenuse of △ABC? (1) The lengths of the three sides of △ABC are consecutive even integers. (2) The

admin2022-10-18  128
问题 What is the length of the hypotenuse of △ABC?
(1) The lengths of the three sides of △ABC are consecutive even integers.
(2) The hypotenuse of AABC is 4 units longer than the shorter leg.
选项 A、Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B、Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C、BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D、EACH statement ALONE is sufficient.
E、Statements (1) and (2) TOGETHER are NOT sufficient.
答案A
解析 (1)    Let n，n + 2, and n + 4 be the consecutive even integers. Using the Pythagorean theorem, we have n2 + (n + 1)2 = (n + 4)2. Because this is a quadratic equation that may have two solutions, we need to investigate further to determine whether there is a unique hypotenuse length.
n2 + (n + 2)2=(n+4)2= (n + 4)2
n2 + n2 + 4n + 4=n2+8n+16
n2 - 4n - 12 =  0
(n-6)(n + 2) =  0
Therefore, n = 6 or n = -2. Since n = -2 corresponds to side lengths of-2,0, and 2, we discard n = -2. Therefore n = 6, the hypotenuse has length n + 4 = 10; SUFFICIENT.
(2)    Let the side lengths be a, b, and a + 4. Using the Pythagorean theorem, we have a2 + b2 = (a + 4)2. Expanding and solving for b in terms of a will facilitate our search for multiple hypotenuse length possibilities.
a2 + b2  =  (a + 4)2
a2+ b2  =  a2 + 8a+16
b2= 8a+16
b =
When a = 1, we obtain side lengths 1 and , and hypotenuse length 5. When a = 2, we obtain
side lengths 2 and , and hypotenuse length 6; NOT sufficient.
The correct answer is A;
statement 1 alone is sufficient.
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