Ratios of Distances between Centroid, Circumcenter, Incenter and Orthocenter of Triangle
Given the ver...
Question
Given the vertices of triangle by position vectors ^i+^j+^k,^i+^kand^j+^k the centroid and Incentre of the triangle will 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 Given position vecto ¯p1,¯p2and¯p3. We know centroid is given by G=p1+p2+p33 I=|¯p2−¯p3|¯p1+|¯p1−¯p3|¯p2+|¯p1−¯p2|¯p3|¯p2−¯p3|+|¯p2−¯p1|+|¯p3−¯p1| Therefor, G=(^i+^j+^k)+(^i+^k)+(^j+^k)3 =2^i+2^j+3^k3 |¯p2−¯p3|=|^i+^k−^j−^k|=|^i−^l|=√2|¯p2−¯p1|=|^i+^k−^i−^j−^k|=|−^j|=1|¯p3−¯p1|=|^j+^k−^i−^j−^k|=|−^i|=1 ∴I=√2(^i+^j+^k)+1(^i+^k)+(^j+^k)√2+1+1 =(√2+1)^i+(√2+1)^j+(√2+2)^k2+√2 Correct option is A.