Difference between revisions of "1954 AHSME Problems/Problem 39"

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Note that the midpoint of <math>P</math> to the point <math>Q</math> is the image of <math>Q</math> under a homothety of factor <math>\frac{1}{2}</math> with center <math>P</math>. Since homotheties preserve circles, the image of the midpoint as <math>Q</math> varies over the circle is a circle centered at the midpoint of <math>P</math> and the original center and radius half the original radius. Therefore, our answer is <math>\boxed{\text{(E)}}</math>, and we are done.
 
Note that the midpoint of <math>P</math> to the point <math>Q</math> is the image of <math>Q</math> under a homothety of factor <math>\frac{1}{2}</math> with center <math>P</math>. Since homotheties preserve circles, the image of the midpoint as <math>Q</math> varies over the circle is a circle centered at the midpoint of <math>P</math> and the original center and radius half the original radius. Therefore, our answer is <math>\boxed{\text{(E)}}</math>, and we are done.
\begin{asy}
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 +
<asy>
 
import graph;
 
import graph;
 
unitsize(60);
 
unitsize(60);
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draw(P--Q, red);
 
draw(P--Q, red);
 
draw(c, darkgreen);
 
draw(c, darkgreen);
label("<math>P</math>", P, N);
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label("$P$", P, N);
label("<math>Q</math>", Q, NW);
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label("$Q$", Q, NW);
  
 
pair M;
 
pair M;
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dot(M, blue);
 
dot(M, blue);
 
draw(Circle(P/2, 1/2, 100), blue);
 
draw(Circle(P/2, 1/2, 100), blue);
label("<math>M</math>", M, SE);
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label("$M$", M, SE);
\end{asy}
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</asy>
  
 
==See Also==
 
==See Also==

Latest revision as of 16:18, 5 July 2020

The locus of the midpoint of a line segment that is drawn from a given external point $P$ to a given circle with center $O$ and radius $r$, is:

$\textbf{(A)}\ \text{a straight line perpendicular to }\overline{PO}\\ \textbf{(B)}\ \text{a straight line parallel to }\overline{PO}\\ \textbf{(C)}\ \text{a circle with center }P\text{ and radius }r\\ \textbf{(D)}\ \text{a circle with center at the midpoint of }\overline{PO}\text{ and radius }2r\\ \textbf{(E)}\ \text{a circle with center at the midpoint }\overline{PO}\text{ and radius }\frac{1}{2}r$

Solution

Note that the midpoint of $P$ to the point $Q$ is the image of $Q$ under a homothety of factor $\frac{1}{2}$ with center $P$. Since homotheties preserve circles, the image of the midpoint as $Q$ varies over the circle is a circle centered at the midpoint of $P$ and the original center and radius half the original radius. Therefore, our answer is $\boxed{\text{(E)}}$, and we are done.

[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("$P$", P, N); label("$Q$", Q, NW);  pair M; M = (P+Q)/2; dot(M, blue); draw(Circle(P/2, 1/2, 100), blue); label("$M$", M, SE); [/asy]

See Also

1954 AHSC (ProblemsAnswer KeyResources)
Preceded by
Problem 35
Followed by
Problem 37
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All AHSME Problems and Solutions


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