question_answer

Points \[(6,\,8),\,(3,7),(-2,-2)\]  and \[(1,-1)\] are joined to form a quadrilateral. What will be the structure of quadrilateral?

A) Rhombus        

B)  Parallelogram

C)         Square       

D)         Rectangle                   

Correct Answer: B

Solution :

Let the points be \[A(6,8),\,\,B(3,\,7),\,\,C(-2,-2)\]And \[D(1,-1)\]             Now,   \[AB=\sqrt{{{(3-6)}^{2}}+{{(7-8)}^{2}}}=\sqrt{10}\] \[BC=\sqrt{{{(-2-3)}^{2}}+{{(-2-7)}^{2}}}=\sqrt{106}\] \[CD=\sqrt{{{(1+2)}^{2}}+{{(-1+2)}^{2}}}=\sqrt{10}\] \[DA=\sqrt{{{(6-1)}^{2}}+{{(8+1)}^{2}}}=\sqrt{106}\] Also\[AC=\sqrt{{{(8)}^{2}}+{{(10)}^{2}}}=\sqrt{64+100}=\sqrt{164}\] \[BD=\sqrt{{{(2)}^{2}}+{{(8)}^{2}}}=\sqrt{4+64}=\sqrt{68}\] Since, \[AB=DC\]and \[BC=DA\]and\[AC\ne BD.\]. \[\therefore \]  It is a parallelogram.