Difference between revisions of "1954 AHSME Problems/Problem 39"
Duck master (talk | contribs) (Created page with solution.) |
Duck master (talk | contribs) m (Fixed Asymptote beginning/end tags.) |
||
Line 6: | Line 6: | ||
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. | ||
− | + | ||
+ | <asy> | ||
import graph; | import graph; | ||
unitsize(60); | unitsize(60); | ||
Line 19: | Line 20: | ||
draw(P--Q, red); | draw(P--Q, red); | ||
draw(c, darkgreen); | draw(c, darkgreen); | ||
− | label(" | + | label("$P$", P, N); |
− | label(" | + | label("$Q$", Q, NW); |
pair M; | pair M; | ||
Line 26: | Line 27: | ||
dot(M, blue); | dot(M, blue); | ||
draw(Circle(P/2, 1/2, 100), blue); | draw(Circle(P/2, 1/2, 100), blue); | ||
− | label(" | + | label("$M$", M, SE); |
− | + | </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 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.
See Also
1954 AHSC (Problems • Answer Key • Resources) | ||
Preceded by Problem 35 |
Followed by Problem 37 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50 | ||
All AHSME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.