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Difference between revisions of "1957 AHSME Problems/Problem 18"

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== Problem 18==
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== Problem ==
  
 
Circle <math>O</math> has diameters <math>AB</math> and <math>CD</math> perpendicular to each other. <math>AM</math> is any chord intersecting <math>CD</math> at <math>P</math>.  
 
Circle <math>O</math> has diameters <math>AB</math> and <math>CD</math> perpendicular to each other. <math>AM</math> is any chord intersecting <math>CD</math> at <math>P</math>.  
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==Solution==
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== Solution 1 ==
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Draw <math>MB</math>. Since <math>\angle AMB</math> is inscribed on a diameter, <math>\angle AMB</math> is <math>90^\circ</math>. By AA Similarity, <math>\triangle APO ~ \triangle ABM</math>. Setting up ratios, we get <math>\frac{AP}{AO}=\frac{AB}{AM}</math>. Cross-multiplying, we get <math>AP\cdot AM = AO \cdot AB</math>, so the answer is <math>\boxed{\textbf{(B)}}</math>.
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Draw <math>MB</math>. Since <math>\angle AMB</math> is inscribed on a diameter, <math>\angle AMB</math> is <math>90^\circ</math>. By [[AA Similarity]], <math>\triangle APO \sim \triangle ABM</math>. Setting up ratios, we get <math>\frac{AP}{AO}=\frac{AB}{AM}</math>. Cross-multiplying, we get <math>AP\cdot AM = AO \cdot AB</math>, so the answer is <math>\boxed{\textbf{(B)}}</math>.
  
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==See Also==
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== Solution 2 ==
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By [[Thales' Theorem]], <math>\measuredangle PMB = \measuredangle AMB = 90^{\circ}</math>. Because, from the problem, <math>\measuredangle POB = 90^{\circ}</math> as well, <math>\measuredangle PMB + \measuredangle POB = 180^{\circ}</math>, so <math>PMBO</math> is a [[cyclic quadrilateral]]. Thus, because <math>P</math>, <math>M</math>, <math>O</math>, and <math>B</math> lie on a circle, we can use [[Power of a Point]]. From this theorem, we get that <math>AP \cdot AM = AO \cdot AB</math>, which is answer choice <math>\boxed{\textbf{(B)}}</math>.
  
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{{AHSME box|year=1957|num-b=17|num-a=19}}
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== See Also ==
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{{AHSME 50p box|year=1957|num-b=17|num-a=19}}
 
{{MAA Notice}}
 
{{MAA Notice}}
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[[Category:Introductory Geometry Problems]]

Latest revision as of 08:53, 25 July 2024

Problem

Circle $O$ has diameters $AB$ and $CD$ perpendicular to each other. $AM$ is any chord intersecting $CD$ at $P$. Then $AP\cdot AM$ is equal to:

[asy] defaultpen(linewidth(.8pt)); unitsize(2cm); pair O = origin; pair A = (-1,0); pair B = (1,0); pair C = (0,1); pair D = (0,-1); pair M = dir(45); pair P = intersectionpoint(O--C,A--M); draw(Circle(O,1)); draw(A--B); draw(C--D); draw(A--M); label("$A$",A,W); label("$B$",B,E); label("$C$",C,N); label("$D$",D,S); label("$M$",M,NE); label("$O$",O,NE); label("$P$",P,NW);[/asy]

$\textbf{(A)}\ AO\cdot OB \qquad \textbf{(B)}\ AO\cdot AB\qquad \\ \textbf{(C)}\ CP\cdot CD \qquad \textbf{(D)}\ CP\cdot PD\qquad \textbf{(E)}\ CO\cdot OP$


Solution 1

Draw $MB$. Since $\angle AMB$ is inscribed on a diameter, $\angle AMB$ is $90^\circ$. By AA Similarity, $\triangle APO \sim \triangle ABM$. Setting up ratios, we get $\frac{AP}{AO}=\frac{AB}{AM}$. Cross-multiplying, we get $AP\cdot AM = AO \cdot AB$, so the answer is $\boxed{\textbf{(B)}}$.

Solution 2

By Thales' Theorem, $\measuredangle PMB = \measuredangle AMB = 90^{\circ}$. Because, from the problem, $\measuredangle POB = 90^{\circ}$ as well, $\measuredangle PMB + \measuredangle POB = 180^{\circ}$, so $PMBO$ is a cyclic quadrilateral. Thus, because $P$, $M$, $O$, and $B$ lie on a circle, we can use Power of a Point. From this theorem, we get that $AP \cdot AM = AO \cdot AB$, which is answer choice $\boxed{\textbf{(B)}}$.


See Also

1957 AHSC (Problems • Answer Key • Resources)
Preceded by
Problem 17 		Followed by
Problem 19
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All AHSME Problems and Solutions

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