AMC 8 1997 Problem 15
AMC8 1997 年第 15 题
四边形与多边形★★★☆☆
Each side of the large square in the figure is trisected (divided into three equal parts). The corners of an inscribed square are at these trisection points, as shown. The ratio of the area of the inscribed square to the area of the large square is
图中大正方形的每条边被三等分。在等分点处作内接正方形，如图所示。内接正方形的面积与大正方形的面积之比是
- A.
- B.
- C.
- D.
- E.
正确答案 · Correct Answer
B
解析 · Solution
Solution 1
Since we are dealing with ratios, let the big square have sides of 3 and thus an area of . Chosing a multiple of 3 will avoid fractions in the rest of the answer.
To find the area of the inscribed square, subtract off the areas of the four triangles. Each triangle has an area of .
Thus, the area of the inscribed square is , and the ratio of areas is , giving option .
Let the side of the big square be . The area of this square is
You can use the Pythagorean Theorem on the any of the right triangles to find the length of the side of the inscribed square. One leg is , while the other leg is . Thus, the length of the hypotenuse , which is also the side of the inscrubed square, is:
, , ,
Once you get , you can stop, because is the area of the inscribed square. The ratio of the areas of the small square to the big square is , giving option .
Fill in the gridlines of the diagram.
Let each grid space have length 1. The inscribed square consists of four congruent triangles and one square.
The triangles each have area , and in total they cover square units of area. The central square's area is . Therefore, the area of the inscribed square is 4+1=5.
The big square contains square units of area. Thus, the ratio is equal to , or .
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