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

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== Problem ==
 
== Problem ==
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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>.
  
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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>.
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== Solution ==
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== Solution 1 ==
 
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From the problem statement, we construct the following diagram:  
 
From the problem statement, we construct the following diagram:  
 
<center><asy>
 
<center><asy>
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Using the [[Pythagorean Theorem]]:
 
Using the [[Pythagorean Theorem]]:
  
−
<cmath>AD^2 = AC^2 + CD^2 </cmath>
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<div style="text-align:center"><math> (AD)^2 = (AC)^2 + (CD)^2 </math></div>
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<cmath>AC^2 = AB^2 + BC^2 </cmath>
+
 
−
Substituting <math>AB^2 + BC^2</math> for <math>AC^2</math>:  
+
<div style="text-align:center"><math> (AC)^2 = (AB)^2 + (BC)^2 </math></div>
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<cmath>AD^2 = AB^2 + BC^2 + CD^2</cmath>
+
 
 +
Substituting <math>(AB)^2 + (BC)^2 </math> for <math> (AC)^2 </math>:  
 +
 
 +
<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:  
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<cmath>AD^2 = 18^2 + 21^2 + 14^2</cmath>
 
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<cmath>AD^2 = 961</cmath>
 
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<cmath>AD= 31</cmath>
 
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So the perimeter is <math>18+21+14+31=84</math>, and the answer is <math>\boxed{084}</math>.
 
  
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== See Also ==
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<div style="text-align:center"><math> (AD)^2 = (18)^2 + (21)^2 + (14)^2 </math></div>
  
 +
<div style="text-align:center"><math> (AD)^2 = 961 </math></div>
 +
 +
<div style="text-align:center"><math> (AD)= 31 </math></div>
 +
 +
So the perimeter is <math> 18+21+14+31=84 </math>, and the answer is <math>\boxed{084}</math>.
 +
 +
== See also ==
 
{{AIME box|year=2006|n=I|before=First Question|num-a=2}}
 
{{AIME box|year=2006|n=I|before=First Question|num-a=2}}
 +
 +
[[Category:Intermediate Geometry Problems]]
 
{{MAA Notice}}
 
{{MAA Notice}}
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[[Category:Introductory Geometry Problems]]
 

Revision as of 13:13, 3 February 2025

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 (Problems • Answer Key • Resources)
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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