Ratios of Distances between Centroid, Circumcenter, Incenter and Orthocenter of Triangle
Given positio...
Question
Given position vectors ¯a,¯b,¯cof points A, B, C for a triangle. The centroid can be given by
A
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B
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C
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D
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Solution
The correct option is A Let’s construct the triangle BC. Let P,M,N be the mid points of sides AB, BC and CA. By section formula P, M and N can be given by ¯a+¯b2,¯b+¯a2and¯c+¯a2. Apart from that we also know the fact that centroid divides the line joining a vertex and midpoint of opposite side in the ratio 2: 3. ∴¯G=¯a+23(¯¯¯¯¯¯¯¯¯¯AM)¯¯¯¯¯¯¯¯¯¯AM=(¯b+¯c2)−¯a⇒¯¯¯¯¯¯¯¯¯¯AM=¯b+¯c−2¯a2⇒¯G=¯a+23(¯b+¯c−2¯a2)=¯a+¯b+¯c3 ∴ Centroid can be given by ¯a+¯b+¯c3