AMC 8 2000 Problem 6
AMC8 2000 年第 6 题
面积与周长（综合）★★☆☆☆
Figure is a square. Inside this square three smaller squares are drawn with the side lengths as labeled. The area of the shaded -shaped region is
ABCD 是个正方形。在这个正方形内画了 3 个小正方形，它们的边长如图所示。则 L 形状的阴影部分区域的面积是多少？
- A.7
- B.10
- C.12.5
- D.14
- E.15
正确答案 · Correct Answer
A
解析 · Solution
Solution 1
The side of the large square is 1 + 3 + 1 = 5, so the area of the large square is . The area of the middle square is , and the sum of the areas of the two smaller squares is . Thus, the big square minus the three smaller squares is 25 - 9 - 2 = 14. This is the area of the two congruent L-shaped regions. So the area of one L-shaped region is , and the answer is
The shaded area can be divided into two regions: one rectangle that is 1 by 3, and one rectangle that is 4 by 1. (Or the reverse, depending on which rectangle the 1 by 1 square is "joined" to.) Either way, the total area of these two regions is 3 + 4 = 7, and the answer is .
Chop the entire 5 by 5 region into 25 squares like a piece of graph paper. When you draw all the lines, you can count that only 7 of the small 1 by 1 squares will be shaded, giving as the answer.
In the bottom left corner of the 5 by 5 square there is a 4 by 4 square which has an area of . In the top right of that 4 by 4 square is a 3 by 3 square with an area of . When we remove the 3 by 3 square from the 4 by 4 square we get the L-shaped figure so our answer is
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