题目信息
What is the length of the hypotenuse of ΔABC ?
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.
参考答案及共享解析
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已选答案：
正确答案：
A:Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Geometry Pythagorean theorem
Let n, n + 2, and n + 4 be the consecutive even integers. Using the Pythagorean theorem, we have n2 + (n + 2)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.
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.
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.
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.
Let n, n + 2, and n + 4 be the consecutive even integers. Using the Pythagorean theorem, we have n2 + (n + 2)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.
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.
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.
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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