we are given
equilateral triangle
we can see that triangle ACD is a right angled triangle
so, we can use Pythagoras theorem for this triangle
hypotenuse=AC
Adjacent=CD
opposite=AD=a
and
[tex]AC=6\sqrt{3}[/tex]
[tex]CD=3\sqrt{3}[/tex]
[tex]AD=a[/tex]
now, we can use Pythagoras theorem
and we get
[tex]AC^2=CD^2+AD^2[/tex]
now, we can plug values
[tex](6\sqrt{3} )^2=(3\sqrt{3} )^2+a^2[/tex]
now, we can solve for a
[tex]108=27+a^2[/tex]
[tex]27+a^2-27=108-27[/tex]
[tex]a^2=81[/tex]
[tex]a=9[/tex]
so, option-A..........Answer
