Difference between revisions of "2006 AIME I Problems/Problem 1"
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== See also == | == See also == | ||
+ | * [[2006 AIME I Problems/Problem 2 | Next problem]] | ||
* [[2006 AIME I Problems]] | * [[2006 AIME I Problems]] | ||
* [[Geometry]] | * [[Geometry]] | ||
[[Category:Intermediate Geometry Problems]] | [[Category:Intermediate Geometry Problems]] |
Revision as of 20:45, 11 March 2007
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:
The answer is 084.