question_answer

The lengths of four sides and a diagonal of the given quadrilateral are indicated in the diagram If A denotes the area and \[l\] the length of the other diagonal, then A and \[l\] are respectively

A)  \[12\sqrt{6},4\sqrt{6}\]                                 

B)  \[12\sqrt{6},5\sqrt{6}\]

C)  \[6\sqrt{6},4\sqrt{6}\] 

D)         \[6\sqrt{6},5\sqrt{6}\]

Correct Answer: A

Solution :

 For \[\Delta \,ABC,\,\,a=6\,cm,\,\,b=5,\,\,c=7\,cm\] \[\therefore \]  \[s=\frac{6+5+7}{2}=9\,cm\] \[\therefore \]  Area of \[\Delta \,ABC=\sqrt{s(s-a)\,(s-b)\,(s-c)}\]                                 \[=\sqrt{9\times (-6)\,(9-5)\,(9-7)}\]                                 \[=\sqrt{9\times 3\times 4\times 2}\]                                 \[=3\times 2\sqrt{6}=6\sqrt{6}\] Thus, area of quadrilateral \[=2\times \] Area of \[\Delta \,ABC\]                                 \[=12\sqrt{6}\,sq\,cm\]                 Again, since \[AE=\frac{1}{2}AD=\frac{l}{2}\] Area of \[\Delta \,ABC=\frac{1}{2}\times BC\times AE\] or            \[6\sqrt{6}=\frac{1}{2}\times 6\times \frac{l}{2}\] or            \[l=4\sqrt{6}\]