Difference between revisions of "1969 AHSME Problems/Problem 10"
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Rockmanex3 (talk | contribs) (Solution to Problem 10) |
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== Solution == | == Solution == | ||
| − | <math>\ | + | <asy> |
| + | draw(circle((0,0),100)); | ||
| + | draw((-300,100)--(300,100),Arrows); | ||
| + | draw((-300,-100)--(300,-100),Arrows); | ||
| + | draw((-300,0)--(300,0),dotted,Arrows); | ||
| + | dot((-200,0)); | ||
| + | dot((0,0)); | ||
| + | dot((200,0)); | ||
| + | draw((-200,100)--(-200,-100),dotted); | ||
| + | draw((200,100)--(200,-100),dotted); | ||
| + | </asy> | ||
| + | The distance between the two parallel tangents is the length of the circle's diameter, so the distance from a point that satisfies the conditions and the two tangents is the length of the circle's radius. From the diagram, there are <math>\boxed{\textbf{(C) } 3}</math> points that satisfies the conditions. | ||
== See also == | == See also == | ||
| − | {{AHSME box|year=1969|num-b=9|num-a=11}} | + | {{AHSME 35p box|year=1969|num-b=9|num-a=11}} |
[[Category: Introductory Geometry Problems]] | [[Category: Introductory Geometry Problems]] | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Latest revision as of 03:01, 7 June 2018
Problem
The number of points equidistant from a circle and two parallel tangents to the circle is:
Solution
![[asy] draw(circle((0,0),100)); draw((-300,100)--(300,100),Arrows); draw((-300,-100)--(300,-100),Arrows); draw((-300,0)--(300,0),dotted,Arrows); dot((-200,0)); dot((0,0)); dot((200,0)); draw((-200,100)--(-200,-100),dotted); draw((200,100)--(200,-100),dotted); [/asy]](http://latex.artofproblemsolving.com/6/2/3/623f96d7742c92cf050fafe22c91dd37e6d8e0d5.png)
The distance between the two parallel tangents is the length of the circle's diameter, so the distance from a point that satisfies the conditions and the two tangents is the length of the circle's radius. From the diagram, there are 
points that satisfies the conditions.
See also
| 1969 AHSC (Problems • Answer Key • Resources) | ||
| Preceded by Problem 9 |
Followed by Problem 11 | |
| 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 | ||
| All AHSME Problems and Solutions | ||
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. 
