Difference between revisions of "1965 AHSME Problems/Problem 4"
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\textbf{(D) }\ 4 \qquad | \textbf{(D) }\ 4 \qquad | ||
\textbf{(E) }\ 8 </math> | \textbf{(E) }\ 8 </math> | ||
| + | |||
| + | |||
| + | == 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, <math>\ell_1 \parallel \ell_3</math>, and <math>\ell_1</math> intersects <math>\ell_2</math>. Therefore, <math>\ell_3</math> also intersects <math>\ell_2</math>. The [[locus]] of all points equidistant from parallel lines <math>\ell_1</math> and <math>\ell_3</math> is a third parallel line in between them. Let this line be <math>\ell_4</math>, and let the distance from <math>\ell_4</math> to either <math>\ell_1</math> or <math>\ell_3</math> be <math>d</math>. The points equidistant from lines <math>\ell_1</math>, <math>\ell_2</math>, and <math>\ell_3</math> must all lie on <math>\ell_4</math> and be a distance <math>d</math> from line <math>\ell_2</math>. There are only 2 points, on either side of <math>\ell_2</math>, which satisfy these conditions. Thus, our answer is <math>\fbox{(C) 2}</math>. | ||
| + | |||
| + | ==See Also== | ||
| + | {{AHSME 40p box|year=1965|num-b=3|num-a=5}} | ||
| + | {{MAA Notice}} | ||
| + | |||
| + | [[Category:Introductory Geometry Problems]] | ||
Latest revision as of 16:04, 18 July 2024
Problem
Line 
intersects line 
and line 
is parallel to . The three lines are distinct and lie in a plane.
The number of points equidistant from all three lines is:
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]](http://latex.artofproblemsolving.com/d/2/c/d2c01b9257450acfe466a965fbbaf216a33bdf2b.png)
The lines are coplanar, 
, and intersects . Therefore, also intersects . The locus of all points equidistant from parallel lines and is a third parallel line in between them. Let this line be 
, and let the distance from to either or be 
. The points equidistant from lines , , and must all lie on and be a distance from line . There are only 2 points, on either side of , which satisfy these conditions. Thus, our answer is 
.
See Also
| 1965 AHSC (Problems • Answer Key • Resources) | ||
| Preceded by Problem 3 |
Followed by Problem 5 | |
| 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 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 | ||
| All AHSME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
