1965 AHSME Problems/Problem 4

Problem

Line $\ell_2$ intersects line $\ell_1$ and line $\ell_3$ is parallel to $\ell_1$ . The three lines are distinct and lie in a plane. The number of points equidistant from all three lines is:

$\textbf{(A)}\ 0 \qquad \textbf{(B) }\ 1 \qquad \textbf{(C) }\ 2 \qquad \textbf{(D) }\ 4 \qquad \textbf{(E) }\ 8$

Solution

$[asy] draw((-36,0)--(36,0), arrow=Arrows); label("$\ell_2$", (40,0)); draw((14,-32)--(30,32), arrow=Arrows); label("$\ell_1$", (32,36)); draw((-4,-32)--(12,32), arrow=Arrows); label("$\ell_3$", (14,36)); draw((5,-32)--(21,32), dotted, arrow=Arrows); label("$\ell_4$", (23,36)); [/asy]$

The lines are coplanar, $\ell_1 \parallel \ell_3$ , and $\ell_1$ intersects $\ell_2$ . Therefore, $\ell_3$ also intersects $\ell_2$ . The locus of all points equidistant from parallel lines $\ell_1$ and $\ell_3$ is a third parallel line in between them. Let this line be $\ell_4$ , and let the distance from $\ell_4$ to either $\ell_1$ or $\ell_3$ be $d$ . The points equidistant from lines $\ell_1$ , $\ell_2$ , and $\ell_3$ must all lie on $\ell_4$ and be a distance $d$ from line $\ell_2$ . There are only 2 points, on either side of $\ell_2$ , which satisfy these conditions. Thus, our answer is $\fbox{(C) 2}$ .

1965 AHSC (Problems • Answer Key • Resources)
Preceded by Problem 3	Followed by Problem 5
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1965 AHSME Problems/Problem 4

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