Question

Please explain it in detail

Answer 1

We will use given values of the sides of trapezium DEFG to find its area.

We know that area of trapezium is given by formula

$Area=\frac{1}{2}(sum \; of \;parallel \;bases)(altitude)$

From picture we can see that parallel bases are EF and DG

where EF=3

$DG=6 \sqrt{3} + 3+ 6 \sqrt{3} = 3+12 \sqrt{3}$

to find altitude we need to use Pythogorean theorem in triangle DEM

$DE^2=EM^2+DM^2$

$12^2=EM^2+(6 \sqrt{3} )^2$

$144=EM^2+108$

$36=EM^2$

$6=EM$

Hence altitude = 6

Now plug these values into formula of area

$Area=\frac{1}{2}(sum \; of \;parallel \;bases)(altitude)$

$Area=\frac{1}{2}(3+3+12 \sqrt{3})(6)$

$Area=\frac{1}{2}(6+12 \sqrt{3})(6)$

$Area=(6+12 \sqrt{3})(3)$

$Area=6(1+2 \sqrt{3})(3)$

$Area=18(1+2 \sqrt{3})$

Hence correct choice is D .