Difference between revisions of "2015 AIME II Problems/Problem 4"
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Call the trapezoid <math>ABCD</math> with <math>AB</math> as the smaller base and <math>CD</math> as the longer. The point where an altitude intersects the larger base be <math>E</math> where <math>E</math> is closer to <math>D</math>. | Call the trapezoid <math>ABCD</math> with <math>AB</math> as the smaller base and <math>CD</math> as the longer. The point where an altitude intersects the larger base be <math>E</math> where <math>E</math> is closer to <math>D</math>. | ||
| − | Subtract the two bases and divide to find that <math>ED</math> is <math>\log 8</math>. The altitude can be expressed as <math>\frac{4}{3} log 8</math>. Therefore, the two legs are <math>\frac{5}{3} \log 8</math>, or <math>\log 32</math>. | + | Subtract the two bases and divide to find that <math>ED</math> is <math>\log 8</math>. The altitude can be expressed as <math>\frac{4}{3} \log 8</math>. Therefore, the two legs are <math>\frac{5}{3} \log 8</math>, or <math>\log 32</math>. |
The perimeter is thus <math>\log 32 + \log 32 + \log 192 + \log 3</math> which is <math>\log 2^{16} 3^2</math>. So <math>p + q = \boxed{018}</math> | The perimeter is thus <math>\log 32 + \log 32 + \log 192 + \log 3</math> which is <math>\log 2^{16} 3^2</math>. So <math>p + q = \boxed{018}</math> | ||
Revision as of 18:16, 27 March 2015
Problem
In an isosceles trapezoid, the parallel bases have lengths 
and 
, and the altitude to these bases has length 
. The perimeter of the trapezoid can be written in the form 
, where 
and 
are positive integers. Find 
.
Solution
Call the trapezoid 
with 
as the smaller base and 
as the longer. The point where an altitude intersects the larger base be 
where is closer to 
.
Subtract the two bases and divide to find that 
is 
. The altitude can be expressed as 
. Therefore, the two legs are 
, or 
.
The perimeter is thus 
which is 
. So 
See also
| 2015 AIME II (Problems • Answer Key • Resources) | ||
| Preceded by Problem 3 |
Followed by Problem 5 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
