Difference between revisions of "1953 AHSME Problems/Problem 14"
Awesome yz (talk | contribs) (→Solution) |
Awesome yz (talk | contribs) (→Solution) |
||
(4 intermediate revisions by the same user not shown) | |||
Line 12: | Line 12: | ||
We will test each option to see if it can be true or not. Links to diagrams are provided. | We will test each option to see if it can be true or not. Links to diagrams are provided. | ||
<cmath>\textbf{(A)}\ p-q\text{ can be equal to }\overline{PQ}</cmath> | <cmath>\textbf{(A)}\ p-q\text{ can be equal to }\overline{PQ}</cmath> | ||
− | Let circle <math>Q</math> be inside circle <math>P</math> and tangent to circle <math>P</math>, and the point of tangency be <math>R</math>. <math>PR = p</math>, and <math>QR = q</math>, so <math>PR - QR = PQ = p-q</math> | + | Let circle <math>Q</math> be inside circle <math>P</math> and tangent to circle <math>P</math>, and the point of tangency be <math>R</math>. <math>PR = p</math>, and <math>QR = q</math>, so <math>PR - QR = PQ = p-q.</math> |
− | + | <asy> pair P, Q, R; P = (0,0); Q = (3,0); R = (4,0); draw(Circle(P,4)); draw(Circle(Q,1)); draw(P--R); dot(P); dot(Q); dot(R); label("$P$",P,S); label("$Q$",Q,S); label("$R$",R,E); </asy> | |
<cmath>\textbf{(B)}\ p+q\text{ can be equal to }\overline{PQ}</cmath> | <cmath>\textbf{(B)}\ p+q\text{ can be equal to }\overline{PQ}</cmath> | ||
− | + | Let circle <math>Q</math> be outside circle <math>P</math> and tangent to circle <math>P</math>, and the point of tangency be <math>R</math>. <math>PR = p</math>, and <math>QR = q</math>, so <math>PR + QR = PQ = p+q.</math> | |
+ | <asy> pair P, Q, R; P = (0,0); Q = (5,0); R = (4,0); draw(Circle(P,4)); draw(Circle(Q,1)); draw(P--Q); dot(P); dot(Q); dot(R); label("$P$",P,S); label("$Q$",Q,S); label("$R$",R,SW); </asy> | ||
<cmath>\textbf{(C)}\ p+q\text{ can be less than }\overline{PQ}</cmath> | <cmath>\textbf{(C)}\ p+q\text{ can be less than }\overline{PQ}</cmath> | ||
− | + | 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> | |
+ | <asy> pair P, Q, R, SS; P = (0,0); Q = (5,0); R = (3,0); SS = (4,0); draw(Circle(P,3)); draw(Circle(Q,1)); draw(P--Q); dot(P); dot(Q); dot(R); dot(SS); label("$P$",P,S); label("$Q$",Q,S); label("$R$",R,SW); label("$S$",SS,SW); </asy> | ||
<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> as shown in the diagram. <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> | |
+ | <asy> pair P, Q, R, SS; P = (0,0); Q = (3,0); R = (6,0); SS = (4.5,0); draw(Circle(P,6)); draw(Circle(Q,1.5)); draw(P--R); dot(P); dot(Q); dot(R); dot(SS); label("$P$",P,S); label("$Q$",Q,S); label("$R$",R,SW); label("$S$",SS,SW); </asy> | ||
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>. | ||
==See Also== | ==See Also== | ||
− | {{AHSME 50p box|year=1953|num-b= | + | {{AHSME 50p box|year=1953|num-b=13|num-a=15}} |
[[Category:Introductory Algebra Problems]] | [[Category:Introductory Algebra Problems]] | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 10:23, 19 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
Let circle
be outside circle
and tangent to circle
, and the point of tangency be
.
, and
, so
Let circle
be outside circle
and not tangent to circle
, and the intersection of
with the circles be
and
respectively.
and
, and
, so
Let circle
be inside circle
and not tangent to circle
, and the intersection of
with the circles be
and
as shown in the diagram.
and
, and
, so
, and
, so
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 13 |
Followed by Problem 15 | |
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.