Difference between revisions of "1965 AHSME Problems/Problem 36"
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== Solution == | == Solution == | ||
| + | |||
| + | <asy> | ||
| + | |||
| + | import geometry; | ||
| + | point O=(0,0); | ||
| + | point A=(10,10); | ||
| + | point B=(10,0); | ||
| + | point C; | ||
| + | line OA=line(O,A); | ||
| + | line OB=line(O,B); | ||
| + | |||
| + | // Lines OA and OB | ||
| + | draw(OA); | ||
| + | draw(OB); | ||
| + | |||
| + | // Points O, A, and B | ||
| + | dot(O); | ||
| + | label("O",O,S); | ||
| + | dot(A); | ||
| + | label("A",A,NW); | ||
| + | dot(B); | ||
| + | label("B",B,S); | ||
| + | |||
| + | // Segments AB and BC | ||
| + | draw(A--B); | ||
| + | pair[] x=intersectionpoints(perpendicular(B,OA),(O--A)); | ||
| + | C=x[0]; | ||
| + | dot(C); | ||
| + | label("C", C, NW); | ||
| + | draw(B--C); | ||
| + | |||
| + | // Right Angle Markers | ||
| + | markscalefactor=0.1; | ||
| + | draw(rightanglemark(O,B,A)); | ||
| + | draw(rightanglemark(B,C,O)); | ||
| + | |||
| + | // Length Labels | ||
| + | label("$a$", midpoint(A--B), E); | ||
| + | label("$b$", midpoint(B--C), NE); | ||
| + | |||
| + | </asy> | ||
| + | |||
<math>\fbox{E}</math> | <math>\fbox{E}</math> | ||
Revision as of 18:26, 19 July 2024
Problem
Given distinct straight lines 
and 
. From a point in a perpendicular is drawn to ;
from the foot of this perpendicular a line is drawn perpendicular to .
From the foot of this second perpendicular a line is drawn perpendicular to ;
and so on indefinitely. The lengths of the first and second perpendiculars are 
and 
, respectively.
Then the sum of the lengths of the perpendiculars approaches a limit as the number of perpendiculars grows beyond all bounds. This limit is:
Solution
![[asy] import geometry; point O=(0,0); point A=(10,10); point B=(10,0); point C; line OA=line(O,A); line OB=line(O,B); // Lines OA and OB draw(OA); draw(OB); // Points O, A, and B dot(O); label("O",O,S); dot(A); label("A",A,NW); dot(B); label("B",B,S); // Segments AB and BC draw(A--B); pair[] x=intersectionpoints(perpendicular(B,OA),(O--A)); C=x[0]; dot(C); label("C", C, NW); draw(B--C); // Right Angle Markers markscalefactor=0.1; draw(rightanglemark(O,B,A)); draw(rightanglemark(B,C,O)); // Length Labels label("$a$", midpoint(A--B), E); label("$b$", midpoint(B--C), NE); [/asy]](http://latex.artofproblemsolving.com/1/e/0/1e08415dd3cbedca63ccb9b3f7b981a38c6bf67e.png)
See Also
| 1965 AHSC (Problems • Answer Key • Resources) | ||
| Preceded by Problem 35 |
Followed by Problem 37 | |
| 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. 
