1974 AMC 12 第 11 题

Question

若

(a,b)

和

(c,d)

是直线

y=mx+k

上的两点，则

(a,b)

与

(c,d)

之间的距离用

a

、

c

和

m

表示为

If

(a,b)

and

(c,d)

are two points on the line whose equation is

y=mx+k,

then the distance between

(a,b)

and

(c,d),

in terms of

a,

c,

and

m,

is

|a-c|\sqrt{1+m^2}

|a+c|\sqrt{1+m^2}

\frac{|a-c|}{\sqrt{1+m^2}}

|a-c|(1+m^2)

|a-c||m|

用直线方程将

b-d

表示成

a-c

的式子

Use the line equation to express

b-d

in terms of

a-c

将

b-d=m(a-c)

代入距离公式

Substitute

b-d=m(a-c)

into the distance formula

因为两点都在该直线上，所以

b-d=m(a-c)

。因此，它们之间的距离为

\begin{aligned} \ell&=\sqrt{(a-c)^2+(b-d)^2}\\ &=\sqrt{(a-c)^2(1+m^2)}\\ &=|a-c|\sqrt{1+m^2} \end{aligned}\text{。}

因此，正确答案是 A。

Since both points are on the line,

b-d=m(a-c).

Their distance is therefore

\begin{aligned} \ell&=\sqrt{(a-c)^2+(b-d)^2}\\ &=\sqrt{(a-c)^2(1+m^2)}\\ &=|a-c|\sqrt{1+m^2}. \end{aligned}

Therefore, the correct answer is A.

其他年份的第 11 题