Difference between revisions of "2006 AIME I Problems/Problem 1"
m (2006 AIME I Problem 1 moved to 2006 AIME I Problems/Problem 1) |
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== Solution == | == Solution == | ||
− | + | From the problem statement, we construct the following diagram: <center>[[Image:Aime06i.1.PNG]]</center> | |
− | + | Using the [[Pythagorean Theorem]]: | |
− | <math> (AC)^2 = (AB)^2 + (BC)^2 </math> | + | <center><math> (AD)^2 = (AC)^2 + (CD)^2 </math></center> |
+ | |||
+ | <center><math> (AC)^2 = (AB)^2 + (BC)^2 </math></center> | ||
Substituting <math>(AB)^2 + (BC)^2 </math> for <math> (AC)^2 </math>: | Substituting <math>(AB)^2 + (BC)^2 </math> for <math> (AC)^2 </math>: | ||
− | <math> (AD)^2 = (AB)^2 + (BC)^2 + (CD)^2 </math> | + | <center><math> (AD)^2 = (AB)^2 + (BC)^2 + (CD)^2 </math></center> |
Plugging in the given information: | Plugging in the given information: | ||
− | <math> (AD)^2 = (18)^2 + (21)^2 + (14)^2 </math> | + | <center><math> (AD)^2 = (18)^2 + (21)^2 + (14)^2 </math></center> |
− | <math> (AD)^2 = 961 </math> | + | <center><math> (AD)^2 = 961 </math></center> |
− | <math> (AD)= 31 </math> | + | <center><math> (AD)= 31 </math></center> |
So the perimeter is: | So the perimeter is: | ||
Line 26: | Line 28: | ||
== See also == | == See also == | ||
* [[2006 AIME I Problems]] | * [[2006 AIME I Problems]] | ||
+ | |||
+ | [[Category:Intermediate Geometry Problems]] |
Revision as of 16:58, 18 July 2006
Problem
In quadrilateral is a right angle, diagonal
is perpendicular to
and
Find the perimeter of
Solution
From the problem statement, we construct the following diagram:
Using the Pythagorean Theorem:
![$(AD)^2 = (AC)^2 + (CD)^2$](http://latex.artofproblemsolving.com/a/7/d/a7d216f23824f0248399bfcf94327646da38392c.png)
![$(AC)^2 = (AB)^2 + (BC)^2$](http://latex.artofproblemsolving.com/0/1/9/0195b55e6d09d7d778973a0c221c3ab651e6db7b.png)
Substituting for
:
![$(AD)^2 = (AB)^2 + (BC)^2 + (CD)^2$](http://latex.artofproblemsolving.com/8/8/7/88701f8a91bf212b3086977771394589ed016aed.png)
Plugging in the given information:
![$(AD)^2 = (18)^2 + (21)^2 + (14)^2$](http://latex.artofproblemsolving.com/f/1/c/f1cdddae8be3cb267c3926faed6cb19a2ee5a96b.png)
![$(AD)^2 = 961$](http://latex.artofproblemsolving.com/0/7/7/0772ff8d7eb763ef832decbf901937b9b64b3181.png)
![$(AD)= 31$](http://latex.artofproblemsolving.com/c/8/7/c87f29dea59845f942599cc4924dc064152c154b.png)
So the perimeter is: