Difference between revisions of "2014 AMC 10A Problems/Problem 22"
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==Solution== | ==Solution== | ||
Note that <math>\tan 15^\circ=\frac{EC}{10} \Rightarrow EC=20-10 \sqrt 3</math>. (If you do not know the tangent half-angle formula, it is <math>\frac{1-\cos a}{\sin a}</math>). Therefore, we have <math>DE=10\sqrt 3</math>. Since <math>ADE</math> is a <math>30-60-90</math> triangle, <math>AE=2 \cdot AD=2 \cdot 10=\boxed{\textbf{(E)} \: 20}</math> | Note that <math>\tan 15^\circ=\frac{EC}{10} \Rightarrow EC=20-10 \sqrt 3</math>. (If you do not know the tangent half-angle formula, it is <math>\frac{1-\cos a}{\sin a}</math>). Therefore, we have <math>DE=10\sqrt 3</math>. Since <math>ADE</math> is a <math>30-60-90</math> triangle, <math>AE=2 \cdot AD=2 \cdot 10=\boxed{\textbf{(E)} \: 20}</math> | ||
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| + | ==Solution 2 (non-trig)== | ||
==See Also== | ==See Also== | ||
Revision as of 13:46, 26 December 2015
Problem
In rectangle 
, 
and 
. Let 
be a point on 
such that 
. What is 
?
Solution
Note that 
. (If you do not know the tangent half-angle formula, it is 
). Therefore, we have 
. Since 
is a 
triangle, 
Solution 2 (non-trig)
See Also
| 2014 AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 21 |
Followed by Problem 23 | |
| 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 10 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
