AMC 8 2022 Problem 24
AMC8 2022 年第 24 题
立体几何★★★★☆
The figure below shows a polygon ABCDEFGH , consisting of rectangles and right triangles. When cut out and folded on the dotted lines, the polygon forms a triangular prism. Suppose that AH = EF = 8 and GH = 14 . What is the volume of the prism?
下图所示的多边形 ABCDEFGH 由长方形和直角三角形组成。当把它剪出并沿虚线折叠后，该多边形可以形成一个底面是三角形的棱柱。假设 AH=EF=8，GH=14。问该棱柱的体积是多少？
- A.112
- B.128
- C.192
- D.240
- E.288
正确答案 · Correct Answer
C
解析 · Solution
解法1
常规解法 · 跟踪折叠对应边
折叠时 贴 、 贴 、 贴 ，得 BJ=CI=8、FG=BC=8、HJ=8。由 GH=14 得 JG=14-8=6。三角形底面 面积 ，柱高 =FG=8，体积 。
巧解 · 先认底面直角三角形
三棱柱底面是直角三角形，两直角边由折叠直接给出：BJ=8、JG=GH-HJ=14-8=6（ 折后 =FG=8）。底面积 ，乘棱长 8 即 192。抓「两直角边 8 和 6、长 8」三件套，免逐边追踪。
Solution 1
Elementary · track the folded edges
Folding sends onto , onto , and onto , giving BJ=CI=8, FG=BC=8, and HJ=8. From GH=14 we get JG=14-8=6. The triangular base has area , and the prism height is FG=8, so the volume is .
Slick · spot the right-triangle base
The prism's base is a right triangle whose legs come straight from the fold: BJ=8 and JG=GH-HJ=14-8=6 (since becomes FG=8). The base area is ; times the length 8 gives 192.
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