Difference between revisions of "Mock AIME 2 2006-2007 Problems/Problem 12"
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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. | 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> | + | <math>[AOD]=\frac{\sqrt{143}}{9}</math> |
Thus <math>[ABCD]=\frac{79\sqrt{143}}{18}\rightarrow\boxed{240}</math> | Thus <math>[ABCD]=\frac{79\sqrt{143}}{18}\rightarrow\boxed{240}</math> |
Revision as of 12:55, 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.
Thus
See also
Problem Source
AoPS users 4everwise and Altheman collaborated to create this problem.