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5.2 · Q37

Q.If the centroid of a tetrahedron OABC is (1,2,−1)(1,2,-1) where A =(a,2,3)=(a,2,3), B=(1,b,2)=(1,b,2), C=(2,1,c)=(2,1,c) respectively, find the distance of P(a,b,c)(a,b,c) from the origin.

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Let O(0,0,0),A(a,2,3),B(1,b,2),C(2,1,c)O(0,0,0),A(a,2,3),B(1,b,2),C(2,1,c), with centroid G(1,2,−1)G(1,2,-1). The tetrahedron centroid formula gives

G=O+A+B+C4.G=\frac{O+A+B+C}{4}.

Component-wise: x: 0+a+1+24=1⇒a+3=4⇒a=1\dfrac{0+a+1+2}{4}=1\Rightarrow a+3=4\Rightarrow a=1. y: 0+2+b+14=2⇒b+3=8⇒b=5\dfrac{0+2+b+1}{4}=2\Rightarrow b+3=8\Rightarrow b=5. z: 0+3+2+c4=−1⇒c+5=−4⇒c=−9\dfrac{0+3+2+c}{4}=-1\Rightarrow c+5=-4\Rightarrow c=-9. …

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