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Difference between revisions of "2012 AMC 8 Problems/Problem 5"

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In the diagram, all angles are right angles and the lengths of the sides are given in centimeters. Note the diagram is not drawn to scale. What is ,<math> X </math> in centimeters?
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==Problem==
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In the diagram, all angles are right angles and the lengths of the sides are given in centimeters. Note the diagram is not drawn to scale. What is , <math> X </math> in centimeters?
  
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[asy]
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<asy>
 
pair A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R;
 
pair A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R;
 
A=(4,0);
 
A=(4,0);
Line 22: Line 23:
 
R=(4,3);
 
R=(4,3);
 
draw(A--B--C--D--E--F--G--H--I--J--K--L--M--N--O--P--Q--R--cycle);
 
draw(A--B--C--D--E--F--G--H--I--J--K--L--M--N--O--P--Q--R--cycle);
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label("<math>X</math>",(3.4,1.5));
+
label("$X$",(3.4,1.5));
 
label("6",(7.6,1.5));
 
label("6",(7.6,1.5));
 
label("1",(7.6,3.5));
 
label("1",(7.6,3.5));
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label("2",(0.8,4.5));
 
label("2",(0.8,4.5));
 
label("2",(1.5,3.8));
 
label("2",(1.5,3.8));
−
label("2",(2.8,2.6));[/asy]
+
label("2",(2.8,2.6));</asy>
  
 
<math> \textbf{(A)}\hspace{.05in}1\qquad\textbf{(B)}\hspace{.05in}2\qquad\textbf{(C)}\hspace{.05in}3\qquad\textbf{(D)}\hspace{.05in}4\qquad\textbf{(E)}\hspace{.05in}5 </math>
 
<math> \textbf{(A)}\hspace{.05in}1\qquad\textbf{(B)}\hspace{.05in}2\qquad\textbf{(C)}\hspace{.05in}3\qquad\textbf{(D)}\hspace{.05in}4\qquad\textbf{(E)}\hspace{.05in}5 </math>
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 +
==Solution 1==
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[[File:2012amc85.png]]
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 +
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<math>1 + 1 + 1 + 2 + X = 1 + 2 + 1 + 6\\
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5 + X = 10\\
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X = 5</math>
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Thus, the answer is <math> \boxed{\textbf{(E)}\ 5} </math>.
 +
 +
==Solution 2==
 +
[[File:bcd57a9d78159bce4d3873f81f5d879beaed1d5a.png|300px|center]]
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 +
Note that we only need to consider the value below the marked red line, so we have the equation:
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<cmath> X + 2 = 6 + 1 </cmath>
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<cmath> X = 5 </cmath>
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 +
Hence, the answer is <math> \boxed{\textbf{(E)}\ 5} </math>.
 +
 +
~[[User:Bloggish|Bloggish]]
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 +
==Video Solution==
 +
https://youtu.be/m4g-Nmot-c8 ~savannahsolver
 +
 +
==See Also==
 +
{{AMC8 box|year=2012|num-b=4|num-a=6}}
 +
{{MAA Notice}}

Latest revision as of 19:16, 3 January 2024

Problem

In the diagram, all angles are right angles and the lengths of the sides are given in centimeters. Note the diagram is not drawn to scale. What is , $X$ in centimeters?

[asy] pair A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R; A=(4,0); B=(7,0); C=(7,4); D=(8,4); E=(8,5); F=(10,5); G=(10,7); H=(7,7); I=(7,8); J=(5,8); K=(5,7); L=(4,7); M=(4,6); N=(0,6); O=(0,5); P=(2,5); Q=(2,3); R=(4,3); draw(A--B--C--D--E--F--G--H--I--J--K--L--M--N--O--P--Q--R--cycle); label("$X$",(3.4,1.5)); label("6",(7.6,1.5)); label("1",(7.6,3.5)); label("1",(8.4,4.6)); label("2",(9.4,4.6)); label("2",(10.4,6)); label("3",(8.4,7.4)); label("1",(7.5,7.8)); label("2",(6,8.5)); label("1",(4.7,7.8)); label("1",(4.3,7.5)); label("1",(3.5,6.5)); label("4",(1.8,6.5)); label("1",(-0.5,5.5)); label("2",(0.8,4.5)); label("2",(1.5,3.8)); label("2",(2.8,2.6));[/asy]

$\textbf{(A)}\hspace{.05in}1\qquad\textbf{(B)}\hspace{.05in}2\qquad\textbf{(C)}\hspace{.05in}3\qquad\textbf{(D)}\hspace{.05in}4\qquad\textbf{(E)}\hspace{.05in}5$

Solution 1

2012amc85.png


$1 + 1 + 1 + 2 + X = 1 + 2 + 1 + 6\\ 5 + X = 10\\ X = 5$

Thus, the answer is $\boxed{\textbf{(E)}\ 5}$.

Solution 2

Bcd57a9d78159bce4d3873f81f5d879beaed1d5a.png

Note that we only need to consider the value below the marked red line, so we have the equation: \[X + 2 = 6 + 1\] \[X = 5\]

Hence, the answer is $\boxed{\textbf{(E)}\ 5}$.

~Bloggish

Video Solution

https://youtu.be/m4g-Nmot-c8 ~savannahsolver

See Also

2012 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 4 		Followed by
Problem 6
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 AJHSME/AMC 8 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

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