A) rectangle
B) square
C) rhombus
D) parallelogram
Correct Answer: B
Solution :
Given complex number are \[P(4+i),Q(1+6i),\] \[R(-4+3i)\]and \[S(-1-2i).\] Now , \[|PQ|=\sqrt{{{(4-1)}^{2}}+{{(1-6)}^{2}}}\] \[=\sqrt{9+25}=\sqrt{34}\] \[|QR|=\sqrt{{{(1+4)}^{2}}+{{(6-3)}^{2}}}\] \[=\sqrt{25+9}=\sqrt{34}\] \[|RS|=\sqrt{{{(-4+1)}^{2}}+{{(3+2)}^{2}}}\] \[=\sqrt{9+25}=\sqrt{34}\] and \[|SP|=\sqrt{{{(1-1-4)}^{2}}+{{(-2-1)}^{2}}}\] \[=\sqrt{25+9}=\sqrt{34}\] \[\therefore \] \[|PQ|=|QR|=|RS|=|SP|\] Now, \[|PR|=\sqrt{{{(4+4)}^{2}}+{{(1-3)}^{3}}}\] \[=\sqrt{64+4}=\sqrt{68}\] and \[|QS|=\sqrt{{{(1+1)}^{2}}+{{(6+2)}^{2}}}\] \[\sqrt{4+64}=\sqrt{68}\] \[\therefore \] \[|PR|=|QS|\] Hence, it is a square.
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