Difference between revisions of "1974 AHSME Problems/Problem 11"

Revision as of 12:43, 5 July 2013

Problem

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

$\mathrm{(A)\ } |a-c|\sqrt{1+m^2} \qquad \mathrm{(B) \ }|a+c|\sqrt{1+m^2} \qquad \mathrm{(C) \ } \frac{|a-c|}{\sqrt{1+m^2}} \qquad$

$\mathrm{(D) \ } |a-c|(1+m^2) \qquad \mathrm{(E) \ }|a-c|\,|m|$

Solution

Notice that, since $(a, b)$ is on $y=mx+k$ , we have $b=am+k$ . Similarly, $d=cm+k$ . Using the distance formula, the distance between the points $(a, b)$ and $(c, d)$ is

$\sqrt{(a-c)^2+(b-d)^2}=\sqrt{(a-c)^2+(am+k-cm-k)^2}$

$=\sqrt{(a-c)^2+m^2(a-c)^2}$

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

And so the answer is $\boxed{\text{A}}$ .

1974 AHSME (Problems • Answer Key • Resources)
Preceded by Problem 10	Followed by Problem 12
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30
All AHSME Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

@@ Line 20: / Line 20: @@
 {{AHSME box|year=1974|num-b=10|num-a=12}}
 [[Category:Introductory Algebra Problems]]
+{{MAA Notice}}

Page

Toolbox

Search

Difference between revisions of "1974 AHSME Problems/Problem 11"

Revision as of 12:43, 5 July 2013

Problem

Solution

See Also