Difference between revisions of "2006 AIME I Problems/Problem 1"

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== Problem ==
 
== Problem ==
In quadrilateral <math> ABCD , \angle B </math> is a right angle, diagonal <math> \overline{AC} </math> is perpendicular to <math> \overline{CD}, AB=18, BC=21, </math> and <math> CD=14. </math> Find the perimeter of <math> ABCD. </math>
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In [[quadrilateral]] <math> ABCD</math>, <math>\angle B </math> is a [[right angle]], [[diagonal]] <math>\overline{AC}</math> is [[perpendicular]] to <math>\overline{CD}</math>, <math>AB=18</math>, <math>BC=21</math>, and <math>CD=14</math>. Find the [[perimeter]] of <math>ABCD</math>.
  
 
== Solution ==
 
== Solution ==
From the problem statement, we construct the following diagram: <center>[[Image:Aime06i.1.PNG]]</center>
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From the problem statement, we construct the following diagram:  
 
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<center><asy>
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pointpen = black; pathpen = black + linewidth(0.65);
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pair C=(0,0), D=(0,-14),A=(-(961-196)^.5,0),B=IP(circle(C,21),circle(A,18));
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D(MP("A",A,W)--MP("B",B,N)--MP("C",C,E)--MP("D",D,E)--A--C); D(rightanglemark(A,C,D,40)); D(rightanglemark(A,B,C,40));
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</asy></center><!-- Asymptote replacement for Image:Aime06i.1.PNG by joml88 -->
 
Using the [[Pythagorean Theorem]]:
 
Using the [[Pythagorean Theorem]]:
  
<center><math> (AD)^2 = (AC)^2 + (CD)^2 </math></center>
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<div style="text-align:center"><math> (AD)^2 = (AC)^2 + (CD)^2 </math></div>
  
<center><math> (AC)^2 = (AB)^2 + (BC)^2 </math></center>
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<div style="text-align:center"><math> (AC)^2 = (AB)^2 + (BC)^2 </math></div>
  
 
Substituting <math>(AB)^2 + (BC)^2 </math> for <math> (AC)^2 </math>:  
 
Substituting <math>(AB)^2 + (BC)^2 </math> for <math> (AC)^2 </math>:  
  
<center><math> (AD)^2 = (AB)^2 + (BC)^2 + (CD)^2 </math></center>
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<div style="text-align:center"><math> (AD)^2 = (AB)^2 + (BC)^2 + (CD)^2 </math></div>
  
 
Plugging in the given information:  
 
Plugging in the given information:  
  
<center><math> (AD)^2 = (18)^2 + (21)^2 + (14)^2 </math></center>
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<div style="text-align:center"><math> (AD)^2 = (18)^2 + (21)^2 + (14)^2 </math></div>
 
 
<center><math> (AD)^2 = 961 </math></center>
 
  
<center><math> (AD)= 31 </math></center>
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<div style="text-align:center"><math> (AD)^2 = 961 </math></div>
  
So the perimeter is:  
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<div style="text-align:center"><math> (AD)= 31 </math></div>
<math> 18+21+14+31=84 </math>
 
  
The answer is 084.
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So the perimeter is <math> 18+21+14+31=84 </math>, and the answer is <math>\boxed{084}</math>.
  
 
== See also ==
 
== See also ==
* [[2006 AIME I Problems]]
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{{AIME box|year=2006|n=I|before=First Question|num-a=2}}
* [[Geometry]]
 
  
 
[[Category:Intermediate Geometry Problems]]
 
[[Category:Intermediate Geometry Problems]]
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{{MAA Notice}}

Latest revision as of 21:43, 2 February 2014

Problem

In quadrilateral $ABCD$, $\angle B$ is a right angle, diagonal $\overline{AC}$ is perpendicular to $\overline{CD}$, $AB=18$, $BC=21$, and $CD=14$. Find the perimeter of $ABCD$.

Solution

From the problem statement, we construct the following diagram:

[asy] pointpen = black; pathpen = black + linewidth(0.65); pair C=(0,0), D=(0,-14),A=(-(961-196)^.5,0),B=IP(circle(C,21),circle(A,18)); D(MP("A",A,W)--MP("B",B,N)--MP("C",C,E)--MP("D",D,E)--A--C); D(rightanglemark(A,C,D,40)); D(rightanglemark(A,B,C,40)); [/asy]

Using the Pythagorean Theorem:

$(AD)^2 = (AC)^2 + (CD)^2$
$(AC)^2 = (AB)^2 + (BC)^2$

Substituting $(AB)^2 + (BC)^2$ for $(AC)^2$:

$(AD)^2 = (AB)^2 + (BC)^2 + (CD)^2$

Plugging in the given information:

$(AD)^2 = (18)^2 + (21)^2 + (14)^2$
$(AD)^2 = 961$
$(AD)= 31$

So the perimeter is $18+21+14+31=84$, and the answer is $\boxed{084}$.

See also

2006 AIME I (ProblemsAnswer KeyResources)
Preceded by
First Question
Followed by
Problem 2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions

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