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Difference between revisions of "2021 Fall AMC 10B Problems/Problem 25"

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A rectangle with side lengths <math>1{ }</math> and <math>3,</math> a square with side length <math>1,</math> and a rectangle <math>R</math> are inscribed inside a larger square as shown. The sum of all possible values for the area of <math>R</math> can be written in the form <math>\tfrac mn</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. What is <math>m+n?</math>
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[asy]
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size(8cm);
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draw((0,0)--(10,0));
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draw((0,0)--(0,10));
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draw((10,0)--(10,10));
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draw((0,10)--(10,10));
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draw((1,6)--(0,9));
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draw((0,9)--(3,10));
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draw((3,10)--(4,7));
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draw((4,7)--(1,6));
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draw((0,3)--(1,6));
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draw((1,6)--(10,3));
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draw((10,3)--(9,0));
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draw((9,0)--(0,3));
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draw((6,13/3)--(10,22/3));
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draw((10,22/3)--(8,10));
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draw((8,10)--(4,7));
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draw((4,7)--(6,13/3));
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label("<math>3</math>",(9/2,3/2),N);
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label("<math>3</math>",(11/2,9/2),S);
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label("<math>1</math>",(1/2,9/2),E);
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label("<math>1</math>",(19/2,3/2),W);
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label("<math>1</math>",(1/2,15/2),E);
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label("<math>1</math>",(3/2,19/2),S);
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label("<math>1</math>",(5/2,13/2),N);
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label("<math>1</math>",(7/2,17/2),W);
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label("<math>R</math>",(7,43/6),W);
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[/asy]
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<math>(\textbf{A})\: 14\qquad(\textbf{B}) \: 23\qquad(\textbf{C}) \: 46\qquad(\textbf{D}) \: 59\qquad(\textbf{E}) \: 67</math>

Revision as of 18:03, 22 November 2021

A rectangle with side lengths $1{ }$ and $3,$ a square with side length $1,$ and a rectangle $R$ are inscribed inside a larger square as shown. The sum of all possible values for the area of $R$ can be written in the form $\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. What is $m+n?$ [asy] size(8cm); draw((0,0)--(10,0)); draw((0,0)--(0,10)); draw((10,0)--(10,10)); draw((0,10)--(10,10)); draw((1,6)--(0,9)); draw((0,9)--(3,10)); draw((3,10)--(4,7)); draw((4,7)--(1,6)); draw((0,3)--(1,6)); draw((1,6)--(10,3)); draw((10,3)--(9,0)); draw((9,0)--(0,3)); draw((6,13/3)--(10,22/3)); draw((10,22/3)--(8,10)); draw((8,10)--(4,7)); draw((4,7)--(6,13/3)); label("$3$",(9/2,3/2),N); label("$3$",(11/2,9/2),S); label("$1$",(1/2,9/2),E); label("$1$",(19/2,3/2),W); label("$1$",(1/2,15/2),E); label("$1$",(3/2,19/2),S); label("$1$",(5/2,13/2),N); label("$1$",(7/2,17/2),W); label("$R$",(7,43/6),W); [/asy] $(\textbf{A})\: 14\qquad(\textbf{B}) \: 23\qquad(\textbf{C}) \: 46\qquad(\textbf{D}) \: 59\qquad(\textbf{E}) \: 67$

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