Dhaka 2014,Junior,P9
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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.
In square $ABCD$,the length of its sides is $5$.$E$,$F$ are two points on $AB$ and $AD$ in such a way so that $\angle ECF=45^{\circ}$.Find the maximum value of the perimeter of $\Delta AEF$.
"Questions we can't answer are far better than answers we can't question"
- Raiyan Jamil
- Posts:138
- Joined:Fri Mar 29, 2013 3:49 pm
Re: Dhaka 2014,Junior,P9
I think the ans is 10.
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Re: Dhaka 2014,Junior,P9
Post the full solution.I have also got the answer $10$.But I couldn't prove that this is the maximum value of the perimeter of $\Delta AEF$.Raiyan Jamil wrote:I think the ans is 10.
"Questions we can't answer are far better than answers we can't question"
- Raiyan Jamil
- Posts:138
- Joined:Fri Mar 29, 2013 3:49 pm
Re: Dhaka 2014,Junior,P9
The perimeter of $\Delta AEF$ is AE+EF+FA. Now, if we try to draw the figure, you will discover that, the smallest perimeter of $\Delta AEF$ will be at that time when AE=AF and $\angle AFE=\angle AEF$ =45 degrees.And it's perimeter will be 8.53553....... Then, if you keep the point C in the centre and move the $\triangle CEF$ clockwise or anticlockwise, you can observe that the area of $\triangle CEF$ will be increasing. And then at one point, the point E and F will be on B and A and the area of $\triangle CEF$ will be (5*5)/2=12.5. At that point, you can observe that, the $\triangle AEF$ will have no area. But it's perimeter will be 5+0+5(AE+EF+FA)=10. By this, you can tell that the highest perimeter will be 10.
But I also have a doubt that if this solution will be suitable and acceptable in the math olympiad.
But I also have a doubt that if this solution will be suitable and acceptable in the math olympiad.
A smile is the best way to get through a tough situation, even if it's a fake smile.