AMC12 2024 A

AMC12 2024 A · Q24

AMC12 2024 A · Q24. It mainly tests Triangles (properties), Area & perimeter.

A $\textit{disphenoid}$ is a tetrahedron whose triangular faces are congruent to one another. What is the least total surface area of a disphenoid whose faces are scalene triangles with integer side lengths?

$\textit{不等四面体}$ 是其三角形面全等四面体。求脸为具有整数边长的不等边三角形的此类不等四面体的最小总表面积。

(A) \sqrt{3} \sqrt{3}

(B) 3\sqrt{15} 3\sqrt{15}

(D) 15\sqrt{7} 15\sqrt{7}

(E) 24\sqrt{6} 24\sqrt{6}

Answer

Correct choice: (D)

正确答案：（D）

Solution

Notice that any scalene $\textit{acute}$ triangle can be the faces of a $\textit{disphenoid}$. (See proof in Solution 2.) As a result, we simply have to find the smallest area a scalene acute triangle with integer side lengths can take on. This occurs with a $4,5,6$ triangle (notice that if you decrease the value of any of the sides the resulting triangle will either be isosceles, degenerate, or non-acute). For this triangle, the semiperimeter is $\frac{15}{2}$, so by Heron’s Formula: A=152⋅72⋅52⋅32=152⋅716=1547 The surface area is simply four times the area of one of the triangles, or $\boxed{\textbf{(D) }15\sqrt{7}}$.

注意到任何不等边$\textit{锐三角形}$ 可以是不等四面体的脸。（见解答 $2$ 中的证明。）因此，我们只需找到具有整数边长的不等边锐三角形的最小面积。这发生在 $4,5,6$ 三角形（注意如果减小任何边，结果三角形将变为等腰、退化或非锐）。对于此三角形，半周长为 $\frac{15}{2}$，由 Heron 公式： $A=\sqrt{15\over2\cdot7\over2\cdot5\over2\cdot3\over2}=\sqrt{15\over16\cdot7}=\sqrt{105\over16}=\frac{\sqrt{105}}{4}$ 表面积只是四个三角形的面积之和，即 $\boxed{\textbf{(D) }\dfrac{\sqrt{105}}{4}}$。

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