Difference between revisions of "2024 AMC 12B Problems/Problem 19"
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<cmath> sin(\theta) + sin(120 - \theta) = \frac{13\sqrt{3}}{14} </cmath> | <cmath> sin(\theta) + sin(120 - \theta) = \frac{13\sqrt{3}}{14} </cmath> | ||
<cmath> sin(\theta) + \frac{ \sqrt{3}}{2}cos( \theta) +\frac{ \sqrt{1}}{2}sin( \theta) = \frac{13\sqrt{3}}{14} </cmath> | <cmath> sin(\theta) + \frac{ \sqrt{3}}{2}cos( \theta) +\frac{ \sqrt{1}}{2}sin( \theta) = \frac{13\sqrt{3}}{14} </cmath> | ||
− | <cmath> \sqrt{3} sin( \theta) + cos( \theta) = \frac{13 }{ | + | <cmath> \sqrt{3} sin( \theta) + cos( \theta) = \frac{13 }{7} </cmath> |
− | <cmath> cos( \theta) = \frac{13 }{ | + | <cmath> cos( \theta) = \frac{13 }{7} - \sqrt{3} sin( \theta) </cmath> |
<cmath> \frac{169 }{49} - \frac{26\sqrt{3} }{7} sin( \theta) + 4 sin( \theta)^2 =0 </cmath> | <cmath> \frac{169 }{49} - \frac{26\sqrt{3} }{7} sin( \theta) + 4 sin( \theta)^2 =0 </cmath> | ||
<cmath> sin( \theta) = \frac{5\sqrt{3} }{14} or \frac{4\sqrt{3} }{7} </cmath> | <cmath> sin( \theta) = \frac{5\sqrt{3} }{14} or \frac{4\sqrt{3} }{7} </cmath> |
Revision as of 14:29, 15 November 2024
Contents
[hide]Problem 19
Equilateral with side length is rotated about its center by angle , where , to form . See the figure. The area of hexagon is . What is ?
Solution 1
let O be circumcenter of the equilateral triangle
OF =
2(area(OFC) + area (OCE)) =
is invalid given
Solution #2
From 's side lengths of 14, we get OF = OC = OE = We let angle FOC = () And therefore angle EOC = 120 - ()
The answer would be 3 * (Area + Area )
Which area = 0.5 *
And area = 0.5 *
Therefore the answer would be 3 * 0.5 * (
Which
So
Therefore
And
Which
can be calculated using addition identity, which gives the answer of
(I would really appreciate if someone can help me fix my code and format)
~mitsuihisashi14 ~luckuso (fixed Latex error )
See also
2024 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 18 |
Followed by Problem 20 |
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 |
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