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

(→‎Solution)
(→‎Solution)
Line 44: Line 44:
 
draw((1,6)--(0,9));
 
draw((1,6)--(0,9));
 
draw((1,6)--(0,6));
 
draw((1,6)--(0,6));
 +
draw((1,6)--(4,6));
 +
draw((4,6)--(4,10));
 
draw((0,9)--(3,10));
 
draw((0,9)--(3,10));
 
draw((3,10)--(4,7));
 
draw((3,10)--(4,7));
Line 69: Line 71:
 
label("$a$",(-1/8,28/3),W);
 
label("$a$",(-1/8,28/3),W);
 
label("$b$",(1,81/8),N);
 
label("$b$",(1,81/8),N);
 +
label("$b$",(3,6),N);
 
</asy>
 
</asy>
  

Revision as of 12:53, 23 November 2021

Problem

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] $(\textbf{A})\: 14\qquad(\textbf{B}) \: 23\qquad(\textbf{C}) \: 46\qquad(\textbf{D}) \: 59\qquad(\textbf{E}) \: 67$


Solution

We see that the polygon bounded by the small square, large square, and rectangle of known lengths is an isosceles triangle. Let’s draw a perpendicular from the vertex of this triangle to its opposing side;

[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((1,6)--(0,6)); draw((1,6)--(4,6)); draw((4,6)--(4,10)); 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); label("$b$",(-1/8,9/2),W); label("$b$",(-1/8,7),W); label("$a$",(1/3,6),N); label("$a$",(-1/8,28/3),W); label("$b$",(1,81/8),N); label("$b$",(3,6),N); [/asy]

Solution in progress ~KingRavi

Video Solution

https://www.youtube.com/watch?v=5mPvkipCvhE


See Also

2021 Fall AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 24 		Followed by
Problem 1
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
All AMC 10 Problems and Solutions

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