Given the vertices of triangle by position vectors -k- -k- and -k- the centroid and Incentre of the triangle will be given byA- 2 -2 -3 k-3- -2-1 -2-1 -2-2 k-2-2B- 2 -2 -3 k-3- -2-2 -2-1 -2-2 k-2 -2C- 2 -2 -3 k-3- -2-1 -2-2 -2-2 k-2-2D- 2 -2 -3 k-3- -2-1 -2-2 -2-1 k-2-2

The correct option is A 2î+2ĵ+3k̂/3,(√(2)+1)î+(√(2)+1)ĵ+(√(2)+2)k̂/2+√(2)Given position vecto p̅_̅1̅,p̅_̅2̅ and p̅_̅3̅. We know centroid is given by G=p_1+p_2+ ...

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Standard XII

Mathematics

Ratios of Distances between Centroid, Circumcenter, Incenter and Orthocenter of Triangle

Given the ver...

Question

Given the vertices of triangle by position vectors $^i +^j +^k,^i +^k a n d^j +^k$ the centroid and Incentre of the triangle will be given by

\frac{2^i + 2^j + 3^k}{3}, \frac{(\sqrt{2} + 1)^i + (\sqrt{2} + 1)^j + (\sqrt{2} + 2)^k}{2 + \sqrt{2}}

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\frac{2^i + 2^j + 3^k}{3}, \frac{(\sqrt{2} + 1)^i + (\sqrt{2} + 2)^j + (\sqrt{2} + 1)^k}{2 + \sqrt{2}}

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\frac{2^i + 2^j + 3^k}{3}, \frac{(\sqrt{2} + 1)^i + (\sqrt{2} + 2)^j + (\sqrt{2} + 2)^k}{2 + \sqrt{2}}

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\frac{2^i + 2^j + 3^k}{3}, \frac{(\sqrt{2} + 2)^i + (\sqrt{2} + 1)^j + (\sqrt{2} + 2)^k}{2 + \sqrt{2}}

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Solution

The correct option is A $\frac{2^i + 2^j + 3^k}{3}, \frac{(\sqrt{2} + 1)^i + (\sqrt{2} + 1)^j + (\sqrt{2} + 2)^k}{2 + \sqrt{2}}$
Given position vecto $_{1},_{2} a n d_{3}$ . We know centroid is given by $G = \frac{p_{1} + p_{2} + p_{3}}{3}$
$I = \frac{|_{2} -_{3} |_{1} + |_{1} -_{3} |_{2} + |_{1} -_{2} |_{3}}{|_{2} -_{3} | + |_{2} -_{1} | + |_{3} -_{1} |}$
Therefor, $G = \frac{(^i +^j +^k) + (^i +^k) + (^j +^k)}{3}$
$= \frac{2^i + 2^j + 3^k}{3}$
$|_{2} -_{3} | = |^i +^k -^j -^k | = |^i -^l | = \sqrt{2} |_{2} -_{1} | = |^i +^k -^i -^j -^k | = | -^j | = 1 |_{3} -_{1} | = |^j +^k -^i -^j -^k | = | -^i | = 1$
$∴ I = \frac{\sqrt{2} (^i +^j +^k) + 1 (^i +^k) + (^j +^k)}{\sqrt{2} + 1 + 1}$
$= \frac{(\sqrt{2} + 1)^i + (\sqrt{2 + 1)}^j + (\sqrt{2} + 2)^k}{2 + \sqrt{2}}$
Correct option is A.

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Given the vertices of triangle by position vectors ^i+^j+^k,^i+^k and ^j+^k the centroid and Incentre of the triangle will be given by

Given the vertices of triangle by position vectors $^i +^j +^k,^i +^k a n d^j +^k$ the centroid and Incentre of the triangle will be given by