Difference between revisions of "2004 AMC 12B Problems/Problem 24"
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== Problem == | == Problem == | ||
| − | In <math>\triangle ABC</math>, <math>AB = BC</math>, and <math>\overline{BD}</math> is an [[altitude]]. Point <math>E</math> is on the extension of <math>\overline{AC}</math> such that <math>BE = 10</math>. The values of <math>\tan \angle CBE</math>, <math>\tan \angle DBE</math>, and <math>\tan \angle ABE</math> form a [[geometric progression]], and the values of <math>\cot \angle DBE, \cot \angle CBE, \cot \angle DBC</math> form an [[arithmetic progression]]. What is the area of <math>\triangle ABC</math>? | + | In <math>\triangle ABC</math>, <math>AB = BC</math>, and <math>\overline{BD}</math> is an [[altitude]]. Point <math>E</math> is on the extension of <math>\overline{AC}</math> such that <math>BE = 10</math>. The values of <math>\tan \angle CBE</math>, <math>\tan \angle DBE</math>, and <math>\tan \angle ABE</math> form a [[geometric progression]], and the values of <math>\cot \angle DBE,</math> <math>\cot \angle CBE,</math> <math>\cot \angle DBC</math> form an [[arithmetic progression]]. What is the area of <math>\triangle ABC</math>? |
| + | |||
| + | <center><asy> | ||
| + | size(120); | ||
| + | defaultpen(0.7); | ||
| + | pair A = (0,0), D = (5*2^.5/3,0), C = (10*2^.5/3,0), B = (5*2^.5/3,5*2^.5), E = (4/3*5*2^0.5,0); | ||
| + | draw(A--D--C--E--B--C--D--B--cycle); | ||
| + | label("\(A\)",A,S); | ||
| + | label("\(B\)",B,N); | ||
| + | label("\(C\)",C,S); | ||
| + | label("\(D\)",D,S); | ||
| + | label("\(E\)",E,S); | ||
| + | </asy></center> | ||
<math>\mathrm{(A)}\ 16 | <math>\mathrm{(A)}\ 16 | ||
Revision as of 10:19, 10 February 2008
Problem
In 
, 
, and 
is an altitude. Point 
is on the extension of 
such that 
. The values of 
, 
, and 
form a geometric progression, and the values of 
form an arithmetic progression. What is the area of ?
![[asy] size(120); defaultpen(0.7); pair A = (0,0), D = (5*2^.5/3,0), C = (10*2^.5/3,0), B = (5*2^.5/3,5*2^.5), E = (4/3*5*2^0.5,0); draw(A--D--C--E--B--C--D--B--cycle); label("\(A\)",A,S); label("\(B\)",B,N); label("\(C\)",C,S); label("\(D\)",D,S); label("\(E\)",E,S); [/asy]](http://latex.artofproblemsolving.com/5/2/f/52feaa38e7d0b32ccce66c425b474ae129e47827.png)
Solution
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See also
| 2004 AMC 12B (Problems • Answer Key • Resources) | |
| Preceded by Problem 23 |
Followed by Problem 25 |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
| All AMC 12 Problems and Solutions | |
