1986 AMC 12 第 29 题

Question

不等边三角形

ABC

的两条高分别长

4

和

12

。若第三条高的长度也是整数，它最大可能是多少？

Two of the altitudes of the scalene triangle

ABC

have length

4

and

12.

If the length of the third altitude is also an integer, what is the biggest it can be?

4

5

6

7

以上均非

none of these

当三角形面积固定时，每条边的长度与其对应的高成反比

For a fixed triangle area, each side is inversely proportional to its corresponding altitude

对与

\frac{1}{4},\frac{1}{12},\frac{1}{h}

成比例的三条边应用三角形不等式

Apply the triangle inequalities to side lengths proportional to

\frac{1}{4},\frac{1}{12},\frac{1}{h}

设第三条高为

h

。由于每条边的长度为公共面积的两倍除以对应的高，三条边的长度成比例于

\frac14,\qquad\frac1{12},\qquad\frac1h\text{。}

两个非平凡的三角形不等式给出

\begin{aligned} \frac1h&\lt\frac14+\frac1{12}=\frac13,\\ \frac1h&\gt\frac14-\frac1{12}=\frac16 \end{aligned}\text{。}

因此

3\lt h\lt6

。其中最大的整数为

5

，而三条高互不相同，符合不等边三角形的要求。

因此正确答案是 B。

Let the third altitude be

h.

Since each side equals twice the common area divided by its altitude, the side lengths are proportional to

\frac14,\qquad\frac1{12},\qquad\frac1h.

The two nontrivial triangle inequalities give

\begin{aligned} \frac1h&\lt\frac14+\frac1{12}=\frac13,\\ \frac1h&\gt\frac14-\frac1{12}=\frac16. \end{aligned}

Thus

3\lt h\lt6.

The largest integral possibility is

5,

and its three altitudes are distinct as required for a scalene triangle.

Therefore the correct answer is B.

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