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The Centroid

If z1, z2, ...

Question

If $z_{1}, z_{2}, z_{3}$ are the vertices of an equilateral triangle ABC such that $| z_{1} - i | = | z_{2} - i | = | z_{3} - i |$ , then $| z_{1} + z_{2} + z_{3} |$ equals to

3 \sqrt{3}

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\sqrt{3}

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3

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\frac{1}{\sqrt{3}}

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Solution

The correct option is D $3$
Therefore,
$z_{1}, z_{2}, z_{3}$ are point equidistant from $i$ .
Hence they lie on a circle centered at $i$ .
Also $z_{1}, z_{2}, z_{3}$ are vertices of an equilateral triangle.
Hence centroid and the circumcentre coincides.
$\frac{z_{1} + z_{2} + z_{3}}{3} = i$
$| z_{1} + z_{2} + z_{3} | = 3 | i |$
$= 3$

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If z1,z2,z3 are the vertices of an equilateral triangle ABC such that |z1−i|=|z2−i|=|z3−i|, then |z1+z2+z3| equals to

The correct option is D 3Therefore,z1,z2,z3 are point equidistant from i.Hence they lie on a circle centered at i.Also z1,z2,z3 are vertices of an equilateral triangle.Hence centroid and the circumcentre coincides.z1+z2+z33=i|z1+z2+z3|=3|i|=3

If $z_{1}, z_{2}, z_{3}$ are the vertices of an equilateral triangle ABC such that $| z_{1} - i | = | z_{2} - i | = | z_{3} - i |$ , then $| z_{1} + z_{2} + z_{3} |$ equals to