Question

# Points z1 and z2 are adjacent vertices of a regular polygon of n sides. Find the vertex z3 adjacent to z2(z3≠z1).

A
z2+(z2z1)[cos2πn±isin2πn].
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B
z2(z1)[cos2πn±isin2πn].
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C
z2+(z2z1)[sin2πn±icos2πn].
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D
z2(z1)[sin2πn±icos2πn].
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Solution

## The correct option is A z2+(z2−z1)[cos2πn±isin2πn].Let C(z0) be the centre of the polygon and A1(z1),A3(z3) be two vertices on either side of A2(z2) as shown in figures I and II.z1,z2 being known to be adjacent and we have to find the vertex z3 in terms of z1 and z2.From fig. (1), rotation being anticlockwise, we havez2−z0=(z1−z0)e2πi/nz3−z0=(z2−z0)e2πi/n. Subtracting we getz3−z2=(z2−z1)e2πi/n∴z3=z2+(z2−z1)e2πi/n ...(1)Similarly, proceeding as above for the second figure in which the rotation is clockwise, we havez3=z2+(z2−z1)e−2πi/n ...(2)∴z3=z2+(z2−z1)e±2πi/n=z2+(z2−z1)[cos2πn±isin2πn].Ans: A

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