Difference between revisions of "2006 AIME A Problems/Problem 1"
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== Solution == | == Solution == | ||
− | + | From the problem statement, we construct the following diagram: <div style="text-align:center">[[Image:Aime06i.1.PNG]]</div> | |
− | [[Image: | ||
− | + | Using the [[Pythagorean Theorem]]: | |
− | + | <div style="text-align:center"><math> (AD)^2 = (AC)^2 + (CD)^2 </math></div> | |
− | + | <div style="text-align:center"><math> (AC)^2 = (AB)^2 + (BC)^2 </math></div> | |
− | |||
− | + | Substituting <math>(AB)^2 + (BC)^2 </math> for <math> (AC)^2 </math>: | |
− | <math> | + | <div style="text-align:center"><math> (AD)^2 = (AB)^2 + (BC)^2 + (CD)^2 </math></div> |
− | + | Plugging in the given information: | |
− | <math> | + | <div style="text-align:center"><math> (AD)^2 = (18)^2 + (21)^2 + (14)^2 </math></div> |
− | <math> | + | <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>084</math>. | ||
== See also == | == See also == |
Revision as of 12:54, 25 September 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:


Substituting for
:

Plugging in the given information:



So the perimeter is , and the answer is
.