AMC 8 1999 Problem 25
AMC8 1999 年第 25 题
三角形★★★★☆
Points , , and are midpoints of the sides of right triangle . Points , , are midpoints of the sides of triangle , etc. If the dividing and shading process is done 100 times (the first three are shown) and AC=CG=6 , then the total area of the shaded triangles is nearest
点 、 和 是直角三角形 各边的中点。点 、、 是三角形 各边的中点，以此类推。如果分割和着色过程进行 100 次（前三次如图所示）且 AC=CG=6，则着色三角形的总面积最接近
- A.6
- B.7
- C.8
- D.9
- E.10
正确答案 · Correct Answer
A
解析 · Solution
解法1
常规解法 · 几何级数求和
直角三角形 两直角边 AC=CG=6，面积 。连各边中点把它分成 4 个全等小三角形，每个面积 ；第一次涂的中点三角形面积 4.5。之后在角上的小三角形 （面积 4.5）里重复，每次所涂三角形缩为上次的 。总面积
巧解 · 无穷等比数列
每轮涂掉当前三角形的 、又进入另 递归，故涂色面积是首项 4.5、公比 的等比数列，和 。涂 100 次与无穷的差极小，故最接近 6。
Solution 1
Elementary · sum the geometric series
Right triangle has legs AC=CG=6, so its area is . Joining midpoints splits it into 4 congruent triangles of area 4.5; the shaded medial triangle has area 4.5. The process repeats inside the corner triangle (also area 4.5), each shaded triangle being of the previous. The total is
Slick · infinite geometric series
Each round shades of the current triangle and recurses into another , so the shaded areas form a geometric series with first term 4.5 and ratio , summing to . After 100 rounds the difference from the infinite sum is negligible, so the total is nearest 6.
继续练习