Difference between revisions of "1957 AHSME Problems/Problem 50"
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== Solution == | == Solution == | ||
+ | |||
+ | <asy> | ||
+ | |||
+ | import geometry; | ||
+ | |||
+ | point A = (0,0); | ||
+ | point B = (16,0); | ||
+ | point O = midpoint(A--B); | ||
+ | point G = (11,0); | ||
+ | point A1, B1, O1; | ||
+ | |||
+ | // Defining A', B', and O' | ||
+ | pair[] a1 = intersectionpoints(circle(A,length(segment(A,G))),perpendicular(A,line(A,B))); | ||
+ | A1 = a1[1]; | ||
+ | pair[] b1 = intersectionpoints(circle(B, length(segment(B,G))),perpendicular(B,line(A,B))); | ||
+ | B1 = b1[1]; | ||
+ | O1 = midpoint(A1--B1); | ||
+ | |||
+ | // Circle w/ Diameter | ||
+ | draw(circle(O,length(segment(A,B))/2)); | ||
+ | draw(A--B); | ||
+ | |||
+ | // Segments AA', BB', A'B', and OO' | ||
+ | draw(A--A1); | ||
+ | draw(B--B1); | ||
+ | draw(A1--B1); | ||
+ | draw(O--O1); | ||
+ | |||
+ | // Points w/ Labels | ||
+ | dot(A); | ||
+ | label("A",A,SW); | ||
+ | dot(B); | ||
+ | label("B",B,SE); | ||
+ | dot(A1); | ||
+ | label("A$'$",A1,NW); | ||
+ | dot(B1); | ||
+ | label("B$'$",B1,NE); | ||
+ | dot(O); | ||
+ | label("O",O,S); | ||
+ | dot(O1); | ||
+ | label("O$'$",O1,N); | ||
+ | dot(G); | ||
+ | label("G",G,S); | ||
+ | |||
+ | // Right angle marks | ||
+ | markscalefactor = 0.11; | ||
+ | draw(rightanglemark(A1,A,B)); | ||
+ | draw(rightanglemark(B1,B,A)); | ||
+ | |||
+ | </asy> | ||
+ | |||
<math>\fbox{\textbf{(B)}remains stationary}</math>. | <math>\fbox{\textbf{(B)}remains stationary}</math>. | ||
Revision as of 16:02, 27 July 2024
Problem
In circle , is a moving point on diameter . is drawn perpendicular to and equal to . is drawn perpendicular to , on the same side of diameter as , and equal to . Let be the midpoint of . Then, as moves from to , point :
Solution
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See Also
1957 AHSC (Problems • Answer Key • Resources) | ||
Preceded by Problem 49 |
Followed by Last Problem | |
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All AHSME Problems and Solutions |
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