BDMO Divisional_2014
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Please don't post problems (by starting a topic) in the "X: Solved" forums. Those forums are only for showcasing the problems for the convenience of the users. You can always post the problems in the main Divisional Math Olympiad forum. Later we shall move that topic with proper formatting, and post in the resource section.
The area of ABC and OBC triangle is 120 and 24 respectively. BC=16, EF=8. Find out the area of OEAF Quadrilateral.
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Re: BDMO Divisional_2014
$EF = \dfrac{1}{2} BC$ so,$EF \parallel BC$
So, $\bigtriangleup AEF = \dfrac{1}{4}\bigtriangleup ABC = 30$
Now, $\bigtriangleup OEF$ is similar to $\bigtriangleup OBC$
So, $\dfrac{\bigtriangleup OEF}{\bigtriangleup OBC} = \dfrac{EF^2}{BC^2} = \dfrac{1}{4}$
$\Rightarrow \bigtriangleup OEF = 6$
SO, area of $OEAF = \bigtriangleup AEF + \bigtriangleup OEF = 30 + 6 = 36$
So, $\bigtriangleup AEF = \dfrac{1}{4}\bigtriangleup ABC = 30$
Now, $\bigtriangleup OEF$ is similar to $\bigtriangleup OBC$
So, $\dfrac{\bigtriangleup OEF}{\bigtriangleup OBC} = \dfrac{EF^2}{BC^2} = \dfrac{1}{4}$
$\Rightarrow \bigtriangleup OEF = 6$
SO, area of $OEAF = \bigtriangleup AEF + \bigtriangleup OEF = 30 + 6 = 36$
The study of mathematics, like the Nile, begins in minuteness but ends in magnificence.
- Charles Caleb Colton
- Charles Caleb Colton
Re: BDMO Divisional_2014
You can't say EF||BC only to see that EF=BC/2. Because E and F mayn't be the midpoints. Thanks