Difference between revisions of "1953 AHSME Problems/Problem 14"
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\textbf{(C)}\ p+q\text{ can be less than }\overline{PQ}\ | \textbf{(C)}\ p+q\text{ can be less than }\overline{PQ}\ | ||
\textbf{(D)}\ p-q\text{ can be less than }\overline{PQ}\ \textbf{(E)}\ \text{none of these} </math> | \textbf{(D)}\ p-q\text{ can be less than }\overline{PQ}\ \textbf{(E)}\ \text{none of these} </math> | ||
+ | |||
+ | ==Solution== | ||
+ | |||
+ | 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> | ||
+ | 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> | ||
+ | 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> | ||
+ | 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> | ||
+ | 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>. | ||
+ | |||
+ | ==See Also== | ||
+ | |||
+ | {{AHSME 50p box|year=1953|num-b=13|num-a=15}} | ||
+ | |||
+ | [[Category:Introductory Algebra Problems]] | ||
+ | {{MAA Notice}} |
Latest revision as of 09:23, 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 | |
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