Difference between revisions of "2021 Fall AMC 10B Problems/Problem 25"
| Line 1: | Line 1: | ||
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> | 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> | ||
| − | + | <asy> | |
size(8cm); | size(8cm); | ||
draw((0,0)--(10,0)); | draw((0,0)--(10,0)); | ||
| Line 18: | Line 18: | ||
draw((8,10)--(4,7)); | draw((8,10)--(4,7)); | ||
draw((4,7)--(6,13/3)); | draw((4,7)--(6,13/3)); | ||
| − | label(" | + | label("$3$",(9/2,3/2),N); |
| − | label(" | + | label("$3$",(11/2,9/2),S); |
| − | label(" | + | label("$1$",(1/2,9/2),E); |
| − | label(" | + | label("$1$",(19/2,3/2),W); |
| − | label(" | + | label("$1$",(1/2,15/2),E); |
| − | label(" | + | label("$1$",(3/2,19/2),S); |
| − | label(" | + | label("$1$",(5/2,13/2),N); |
| − | label(" | + | label("$1$",(7/2,17/2),W); |
| − | label(" | + | label("$R$",(7,43/6),W); |
| − | + | </asy> | |
<math>(\textbf{A})\: 14\qquad(\textbf{B}) \: 23\qquad(\textbf{C}) \: 46\qquad(\textbf{D}) \: 59\qquad(\textbf{E}) \: 67</math> | <math>(\textbf{A})\: 14\qquad(\textbf{B}) \: 23\qquad(\textbf{C}) \: 46\qquad(\textbf{D}) \: 59\qquad(\textbf{E}) \: 67</math> | ||
Revision as of 18:32, 22 November 2021
A rectangle with side lengths 
and 
a square with side length 
and a rectangle 
are inscribed inside a larger square as shown. The sum of all possible values for the area of can be written in the form 
, where 
and 
are relatively prime positive integers. What is 
![[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]](http://latex.artofproblemsolving.com/7/3/d/73d3dad024e36cb7991fb09eac46f40b460c89a8.png)
