1954 AHSME Problems/Problem 39
The locus of the midpoint of a line segment that is drawn from a given external point to a given circle with center and radius , is:
Solution
Note that the midpoint of to the point is the image of under a homothety of factor with center . Since homotheties preserve circles, the image of the midpoint as varies over the circle is a circle centered at the midpoint of and the original center and radius half the original radius. Therefore, our answer is , and we are done. \begin{asy} import graph; unitsize(60);
pair P, Q; path c; P = (2,0); Q = dir(142); c = Circle((0,0), 1, 100); dot(P, red); dot(Q, darkgreen); draw(P--Q, red); draw(c, darkgreen); label("", P, N); label("", Q, NW);
pair M; M = (P+Q)/2; dot(M, blue); draw(Circle(P/2, 1/2, 100), blue); label("", M, SE); \end{asy}
See Also
1954 AHSC (Problems • Answer Key • Resources) | ||
Preceded by Problem 35 |
Followed by Problem 37 | |
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