Difference between revisions of "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]]</ | + | From the problem statement, we construct the following diagram: <div style="text-align:center">[[Image:Aime06i.1.PNG]]</div> |
Using the [[Pythagorean Theorem]]: | Using the [[Pythagorean Theorem]]: | ||
− | <center><math> (AD)^2 = (AC)^2 + (CD)^2 </math></ | + | <div style="text-align:center"><math> (AD)^2 = (AC)^2 + (CD)^2 </math></div> |
− | <center><math> (AC)^2 = (AB)^2 + (BC)^2 </math></ | + | <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>: | Substituting <math>(AB)^2 + (BC)^2 </math> for <math> (AC)^2 </math>: | ||
− | <center><math> (AD)^2 = (AB)^2 + (BC)^2 + (CD)^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: | ||
− | <center><math> (AD)^2 = (18)^2 + (21)^2 + (14)^2 </math></ | + | <div style="text-align:center"><math> (AD)^2 = (18)^2 + (21)^2 + (14)^2 </math></div> |
− | <center><math> (AD)^2 = 961 </math></ | + | <div style="text-align:center"><math> (AD)^2 = 961 </math></div> |
− | <center><math> (AD)= 31 </math></ | + | <div style="text-align:center"><math> (AD)= 31 </math></div> |
− | So the perimeter is | + | So the perimeter is <math> 18+21+14+31=84 </math>, and the answer is <math>084</math>. |
− | <math> 18+21+14+31=84 </math> | ||
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== See also == | == See also == |
Revision as of 21:24, 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:
Substituting for :
Plugging in the given information:
So the perimeter is , and the answer is .
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 |