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2006 AIME I Problems/Problem 1: Difference between revisions

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== Solution 1 ==
== Solution 1 ==


From the problem statement, we construct the following diagram:  
We construct the following diagram:  
<center><asy>
<asy>
pointpen = black; pathpen = black + linewidth(0.65);
pathpen = black;
pair C=(0,0), D=(0,-14),A=(-(961-196)^.5,0),B=IP(circle(C,21),circle(A,18));
pair C=(0,0),D=(0,-14),A=(-sqrt(765),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));
D(MP("A",A,W)--MP("B",B,N)--MP("C",C,E)--MP("D",D,E)--A--C);
</asy></center><!-- Asymptote replacement for Image:Aime06i.1.PNG by joml88 -->
D(rightanglemark(A,C,D,40));
Using the [[Pythagorean Theorem]]:
D(rightanglemark(A,B,C,40));
 
</asy><!--Asymptote by joml88-->
<cmath>AD^2 = AC^2 + CD^2 </cmath>
Using the [[Pythagorean Theorem]], we get the following two equations:
<cmath>AC^2 = AB^2 + BC^2 </cmath>
<cmath>AD^2 = AC^2 + CD^2</cmath>
Substituting <math>AB^2 + BC^2</math> for <math>AC^2</math>:
<cmath>AC^2 = AB^2 + BC^2</cmath>
<cmath>AD^2 = AB^2 + BC^2 + CD^2</cmath>
Substituting <math>AB^2 + BC^2</math> for <math>AC^2</math> gives us <math>AD^2 = AB^2 + BC^2 + CD^2</math>. Plugging in the given information, we get <math>AD^2 = 18^2 + 21^2 + 14^2 = 961 \implies AD = 31</math>, so the perimeter is <math>AB+BC+CD+AD = 18+21+14+31 = \boxed{84}</math>.
Plugging in the given information:
<cmath>AD^2 = 18^2 + 21^2 + 14^2</cmath>
<cmath>AD^2 = 961</cmath>
<cmath>AD= 31</cmath>
So the perimeter is <math>18+21+14+31=84</math>, and the answer is <math>\boxed{084}</math>.


== See Also ==
== See Also ==

Latest revision as of 06:22, 30 August 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 1

We construct the following diagram: [asy] pathpen = black; pair C=(0,0),D=(0,-14),A=(-sqrt(765),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, we get the following two equations: \[AD^2 = AC^2 + CD^2\] \[AC^2 = AB^2 + BC^2\] Substituting $AB^2 + BC^2$ for $AC^2$ gives us $AD^2 = AB^2 + BC^2 + CD^2$. Plugging in the given information, we get $AD^2 = 18^2 + 21^2 + 14^2 = 961 \implies AD = 31$, so the perimeter is $AB+BC+CD+AD = 18+21+14+31 = \boxed{84}$.

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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