Difference between revisions of "2007 AMC 12B Problems/Problem 14"
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<cmath>s = \boxed{\mathrm{(D) \ } 4\sqrt{3}}</cmath> | <cmath>s = \boxed{\mathrm{(D) \ } 4\sqrt{3}}</cmath> | ||
| − | *Note - This is called Viviani's Theorem | + | *Note - This is called [[Viviani's Theorem]]. |
==See Also== | ==See Also== | ||
Revision as of 13:56, 31 January 2021
- The following problem is from both the 2007 AMC 12B #14 and 2007 AMC 10B #17, so both problems redirect to this page.
Problem 14
Point 
is inside equilateral 
. Points 
, 
, and 
are the feet of the perpendiculars from to 
, 
, and 
, respectively. Given that 
, 
, and 
, what is 
?
Solution
Drawing 
, 
, and 
, is split into three smaller triangles. The altitudes of these triangles are given in the problem as 
, 
, and 
.
Summing the areas of each of these triangles and equating it to the area of the entire triangle, we get:
![\[\frac{s}{2} + \frac{2s}{2} + \frac{3s}{2} = \frac{s^2\sqrt{3}}{4}\]](http://latex.artofproblemsolving.com/6/b/a/6bacba6cae717b6b1e81eb82d97b264b996dd17c.png)
where 
is the length of a side
![\[s = \boxed{\mathrm{(D) \ } 4\sqrt{3}}\]](http://latex.artofproblemsolving.com/d/d/d/ddd17ba78b8e87e6d688b7af272364fd4811013b.png)
- Note - This is called Viviani's Theorem.
See Also
| 2007 AMC 12B (Problems • Answer Key • Resources) | |
| Preceded by Problem 13 |
Followed by Problem 15 |
| 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 | |
| 2007 AMC 10B (Problems • Answer Key • Resources) | ||
| Preceded by Problem 16 |
Followed by Problem 18 | |
| 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. 
