AMC8 2022

AMC8 2022 · Q24

AMC8 2022 · Q24. It mainly tests Area & perimeter, 3D geometry (volume).

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?

下面的图形显示了一个由矩形和直角三角形组成的八边形$ABCDEFGH$。沿虚线剪下并折叠后，形成一个三棱柱。已知$AH = EF = 8$且$GH = 14$。该棱柱的体积是多少？

(A) 112 112

(B) 128 128

(D) 240 240

(E) 288 288

Answer

Correct choice: (C)

正确答案：（C）

Solution

While imagining the folding, $\overline{AB}$ goes on $\overline{BC},$ $\overline{AH}$ goes on $\overline{CI},$ and $\overline{EF}$ goes on $\overline{FG}.$ So, $BJ=CI=8$ and $FG=BC=8.$ Also, $\overline{HJ}$ becomes an edge parallel to $\overline{FG},$ so that means $HJ=8.$ Since $GH=14,$ then $JG=14-8=6.$ So, the area of $\triangle BJG$ is $\frac{8\cdot6}{2}=24.$ If we let $\triangle BJG$ be the base, then the height is $FG=8.$ So, the volume is $24\cdot8=\boxed{\textbf{(C)} ~192}.$

想象折叠过程，$\overline{AB}$贴$\overline{BC}$，$\overline{AH}$贴$\overline{CI}$，$\overline{EF}$贴$\overline{FG}$。所以，$BJ=CI=8$且$FG=BC=8$。另外，$\overline{HJ}$成为平行于$\overline{FG}$的棱，故$HJ=8$。因为$GH=14$，则$JG=14-8=6$。$\triangle BJG$面积为$\frac{8\cdot6}{2}=24$。以$\triangle BJG$为底，高为$FG=8$，体积$24\cdot8=\boxed{\textbf{(C)} ~192}$。

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