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

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== Problem ==
 
== Problem ==
In [[convex polygon|convex]] [[hexagon]] <math>ABCDEF</math>, all six sides are congruent, <math>\angle A</math> and <math>\angle D</math> are [[right angle]]s, and <math>\angle B, \angle C, \angle E,</math> and <math>\angle F</math> are [[congruent]]. The area of the hexagonal region is <math>2116(\sqrt{2}+1).</math> Find <math>AB</math>.
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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>
  
 
== Solution ==
 
== Solution ==
Let the side length be called <math>x</math>, so <math>x=AB=BC=CD=DE=EF=AF</math>.
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From the problem statement, we construct the following diagram: <div style="text-align:center">[[Image:Aime06i.1.PNG]]</div>
  
[[Image:2006_I_AIME-1.png]]
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Using the [[Pythagorean Theorem]]:
  
The diagonal <math>BF=\sqrt{AB^2+AF^2}=\sqrt{x^2+x^2}=x\sqrt{2}</math>. Then the areas of the triangles AFB and CDE in total are <math>\frac{x^2}{2}\cdot 2</math>,
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<div style="text-align:center"><math> (AD)^2 = (AC)^2 + (CD)^2 </math></div>
and the area of the rectangle BCEF equals <math>x\cdot x\sqrt{2}=x^2\sqrt{2}</math>
 
  
Then we have to solve the equation
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<div style="text-align:center"><math> (AC)^2 = (AB)^2 + (BC)^2 </math></div>
<div style="text-align:center;">
 
<math>2116(\sqrt{2}+1)=x^2\sqrt{2}+x^2</math>.
 
  
<math>2116(\sqrt{2}+1)=x^2(\sqrt{2}+1)</math>
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Substituting <math>(AB)^2 + (BC)^2 </math> for <math> (AC)^2 </math>:
  
<math>2116=x^2</math>
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<div style="text-align:center"><math> (AD)^2 = (AB)^2 + (BC)^2 + (CD)^2 </math></div>
  
<math>x=46</math></div>
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Plugging in the given information:
  
Therefore, <math>AB</math> is <math>046</math>.
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<div style="text-align:center"><math> (AD)^2 = (18)^2 + (21)^2 + (14)^2 </math></div>
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<div style="text-align:center"><math> (AD)^2 = 961 </math></div>
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<div style="text-align:center"><math> (AD)= 31 </math></div>
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So the perimeter is <math> 18+21+14+31=84 </math>, and the answer is <math>084</math>.
  
 
== See also ==
 
== See also ==

Revision as of 19:21, 25 September 2007

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:

Aime06i.1.PNG

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 $084$.

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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