AMC 8 1995 Problem 9
AMC8 1995 年第 9 题
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Three congruent circles with centers , , and are tangent to the sides of rectangle as shown. The circle centered at has diameter 4 and passes through points and . The area of the rectangle is
三个全等的圆，圆心分别为 、 和 ，如图所示与矩形 的边相切。以 为圆心的圆直径为 4 且经过点 和 。矩形的面积是
- A.16
- B.24
- C.32
- D.64
- E.128
正确答案 · Correct Answer
C
解析 · Solution
Solution 1
If circle has diameter 4, then so do congruent circles and . Draw a diameter through parallel to . The diameter will be congruent to , and thus AD = 4, which is the height of the rectangle.
Draw a horizontal line that extends to the sides of the rectangle. This line is 2 diameters long, so it has length . It is parallel and congruent to , so the width of the rectangle is 8.
Thus, the area of the rectangle is , and the answer is .
The area of circle and circle almost fill the rectangle. Circle has radius , and so does circle . Thus, the sum of their areas is . Since , the area of the circles is just over 24. The area of the rectangle is greater than the area of the circles. 64 would be too high, as the two circles appear to take up much more than half the area of the rectangle. But 24 is too low, as it is less than the area of the two circles. Thus, the only reasonable answer is , which is 32.
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