1974 AMC 12 Problem 11

Attempt Problem 11 of the 1974 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1974 AMC 12 solutions, or check the answer key.

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11.

(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,

|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|

Use the line equation to express

b-d

in terms of

a-c

Substitute

b-d=m(a-c)

into the distance formula

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.

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