1990 AMC 12 第 7 题

Question

一个边长均为整数的三角形，其周长为

8

。这个三角形的面积是

A triangle with integral sides has perimeter

8.

The area of the triangle is

2\sqrt2

\frac{16}{9}\sqrt3

2\sqrt3

4

4\sqrt2

列出和为

8

的所有无序正整数三元组

List the unordered triples of positive integers summing to

8

只有一个三元组满足严格的三角不等式

Only one triple satisfies the strict triangle inequality

和为

8

且满足三角不等式的无序正整数边长只有

2,3,3

。半周长为

4

，因此海伦公式给出

\begin{aligned} K&=\sqrt{4(4-2)(4-3)(4-3)}\\ &=2\sqrt2 \end{aligned}\text{。}

所以正确答案是 A。

The only unordered positive integral side lengths summing to

8

and satisfying the triangle inequality are

2,3,3.

Their semiperimeter is

4,

so Heron’s formula gives

\begin{aligned} K&=\sqrt{4(4-2)(4-3)(4-3)}\\ &=2\sqrt2. \end{aligned}

Thus the correct answer is A.

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