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1957 AHSME Problems/Problem 50

Revision as of 17:02, 27 July 2024 by Thepowerful456 (talk | contribs) (diagram)

Problem

In circle $O$, $G$ is a moving point on diameter $\overline{AB}$. $\overline{AA'}$ is drawn perpendicular to $\overline{AB}$ and equal to $\overline{AG}$. $\overline{BB'}$ is drawn perpendicular to $\overline{AB}$, on the same side of diameter $\overline{AB}$ as $\overline{AA'}$, and equal to $BG$. Let $O'$ be the midpoint of $\overline{A'B'}$. Then, as $G$ moves from $A$ to $B$, point $O'$:

$\textbf{(A)}\ \text{moves on a straight line parallel to }{AB}\qquad \\ \textbf{(B)}\ \text{remains stationary}\qquad\\ \textbf{(C)}\ \text{moves on a straight line perpendicular to }{AB}\qquad\\ \textbf{(D)}\ \text{moves in a small circle intersecting the given circle}\qquad\\ \textbf{(E)}\ \text{follows a path which is neither a circle nor a straight line}$

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]

$\fbox{\textbf{(B)}remains stationary}$.

See Also

1957 AHSC (Problems • Answer Key • Resources)
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