Difference between revisions of "Mock AIME 2 2006-2007 Problems/Problem 12"
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==Solution== | ==Solution== | ||
− | {{ | + | <math>m\angle DAC=m\angle DBC \Rightarrow ABCD</math> is a cylic quadrilateral. |
+ | |||
+ | Let <math>DO=a, AO=b</math> | ||
+ | |||
+ | <math>\triangle AOD</math> ~ <math>\triangle BOC \Rightarrow b=\frac{2}{3}</math> | ||
+ | |||
+ | Also, from the Power of a Point Theorem, <math>DO \cdot BO=AO\cdot CO\Rightarrow CO=\frac{3a}{2}</math> | ||
+ | |||
+ | Notice <math>\sin{\angle AOD}=\sin{(180-\angle AOD)}</math> , <math>[AOD]=\sin{\angle AOD}\cdot\frac{1}{2}\cdot a\cdot\frac{2}{3}=\frac{a}{3}\cdot\sin{\angle AOD}</math> , <math>[AOB]=\sin{(180-\angle AOD)}\cdot\frac{1}{2}\cdot\frac{2}{3}\cdot 1=\frac{[AOD]}{a}</math> | ||
+ | |||
+ | It is given <math>\frac{[AOD]+[AOB]}{[AOB]+[BOC]}=\frac{[ADB]}{[ABC]}=\frac{1}{2} \Rightarrow</math> | ||
+ | |||
+ | <math>[AOB]=\frac{[AOD]}{4}</math> | ||
+ | |||
+ | <math>[BOC]=\frac{9}{4}[AOD]</math> | ||
+ | |||
+ | <math>[COD]=36[AOD]</math> | ||
+ | |||
+ | <math>\Rightarrow a=4</math> | ||
+ | |||
+ | Thus we need to find <math>[ABCD]=\frac{79}{2}[AOD]</math> | ||
+ | |||
+ | Note that <math>\triangle AOD</math> is isosceles with sides <math>4, 4, \frac{2}{3}</math> so we can draw the altitude from D to split it to two right triangles. | ||
+ | |||
+ | <math>[AOD]=\frac{\sqrt{143}{9}</math> | ||
+ | |||
+ | Thus <math>[ABCD]=\frac{79\sqrt{143}}{18}\rightarrow\boxed{240}</math> | ||
+ | |||
+ | |||
==See also== | ==See also== |
Revision as of 12:22, 24 February 2009
Contents
Problem
In quadrilateral and is defined to be the intersection of the diagonals of . If , and the area of is where are relatively prime positive integers, find
Note*: and refer to the areas of triangles and
Solution
is a cylic quadrilateral.
Let
~
Also, from the Power of a Point Theorem,
Notice , ,
It is given
Thus we need to find
Note that is isosceles with sides so we can draw the altitude from D to split it to two right triangles.
$[AOD]=\frac{\sqrt{143}{9}$ (Error compiling LaTeX. Unknown error_msg)
Thus
See also
Problem Source
AoPS users 4everwise and Altheman collaborated to create this problem.