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O≡(0,0), A≡(2,0)AND B≡(1,√3). P(x,y) |
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| Aug16-09, 07:19 AM | #1 |
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O≡(0,0), A≡(2,0)AND B≡(1,√3). P(x,y)
consider a triangle OAB formed by O≡(0,0), A≡(2,0)AND B≡(1,√3). P(x,y) is an arbitrary interior point of the triangle moving in such a way that the sum of its distances from the three sides of the triangle is √3 units . find the area of the region representing possible positions of the point P .
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| Aug16-09, 09:33 AM | #2 |
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Hi abhishek!
 (this is an equilateral triangle, of course, and one of the possible positions of P is its centroid) Show us what you've tried, and where you're stuck, and then we'll know how to help! 
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| Aug27-09, 01:38 PM | #3 |
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Spoiler
the triangle is an equilateral triangle, so the distance from any arbitrary point P inside the triangle to the sides of the triangle is constant (and equal to the height of the triangle, so √3 units in this case).
proof: consider P, an arbitrary point inside the triangle.  the area of the triangle OAB is √3 units and it is also the sum of the areas of the triangles POA= (d1*0A)/2, POB=(d2*OB)/2 and PAB=(d3*AB)/2. but OA=OB=AB=2 units. So the sum of the areas of the smaller triangles is d1+d2+d3=√3. Hence, the sum of the distances from P to the sides of the triangle is constant and equal to √3. q.e.d. so the area of the region representing possible positions of the point P that satisfy the given positions is the area of the triangle = √3 square units... |
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