Difference between revisions of "1953 AHSME Problems/Problem 14"
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Let circle <math>Q</math> be outside circle <math>P</math> and not tangent to circle <math>P</math>, and the intersection of <math>\overline{PQ}</math> with the circles be <math>R</math> and <math>S</math> respectively. <math>PR = p</math> and <math>QS = q</math>, and <math>PR + QS < PQ</math>, so <math>p+q < PQ.</math> [https://latex.artofproblemsolving.com/2/0/d/20d001912b80a7a6ef4c267abdd424da9ca14784.png Diagram C] | Let circle <math>Q</math> be outside circle <math>P</math> and not tangent to circle <math>P</math>, and the intersection of <math>\overline{PQ}</math> with the circles be <math>R</math> and <math>S</math> respectively. <math>PR = p</math> and <math>QS = q</math>, and <math>PR + QS < PQ</math>, so <math>p+q < PQ.</math> [https://latex.artofproblemsolving.com/2/0/d/20d001912b80a7a6ef4c267abdd424da9ca14784.png Diagram C] | ||
<cmath>\textbf{(D)}\ p-q\text{ can be less than }\overline{PQ}</cmath> | <cmath>\textbf{(D)}\ p-q\text{ can be less than }\overline{PQ}</cmath> | ||
− | + | Let circle <math>Q</math> be inside circle <math>P</math> and not tangent to circle <math>P</math>, and the intersection of <math>\overline{PQ}</math> with the circles be <math>R</math> and <math>S</math> respectively. <math>PR = p</math> and <math>QS = q</math>, and <math>QS < QR</math>, so <math>PR - QS < PR - QR</math>, and <math>PR - QR = PQ</math>, so <math>p-q < PQ.</math> | |
+ | [https://latex.artofproblemsolving.com/2/2/9/229380edd13828236175da31534b31fe96f3b47b.png Diagram D] | ||
Since options A, B, C, and D can be true, the answer must be <math>\boxed{E}</math>. | Since options A, B, C, and D can be true, the answer must be <math>\boxed{E}</math>. | ||
Revision as of 15:10, 15 July 2018
Problem 14
Given the larger of two circles with center and radius
and the smaller with center
and radius
. Draw
. Which of the following statements is false?
Solution
We will test each option to see if it can be true or not. Links to diagrams are provided.
Let circle
be inside circle
and tangent to circle
, and the point of tangency be
.
, and
, so
Diagram A
Let circle
be outside circle
and tangent to circle
, and the point of tangency be
.
, and
, so
Diagram B
Let circle
be outside circle
and not tangent to circle
, and the intersection of
with the circles be
and
respectively.
and
, and
, so
Diagram C
Let circle
be inside circle
and not tangent to circle
, and the intersection of
with the circles be
and
respectively.
and
, and
, so
, and
, so
Diagram D
Since options A, B, C, and D can be true, the answer must be
.
See Also
1953 AHSC (Problems • Answer Key • Resources) | ||
Preceded by Problem 12 |
Followed by Problem 14 | |
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.