1965 AHSME Problems/Problem 36
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 | |
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| All AHSME Problems and Solutions | ||
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