in equilateral ΔABC, AD, BE, and CF are medians. If FO = 4, then AE = A) 4 B) 4 3 C) 8 D) 12

Firstly, we need to draw triangle we know that O is a centroid and centroid divides median into 2:1so, [tex]\frac{CO}{FO} =\frac{2}{1}[/tex]we have FO=4so, we can plug it [tex]\frac{CO}{4} =\frac{2}{1}[/tex][tex]CO=8[/tex]now, we can find CFCF=OC+FOCF=8+4CF=12now, we can see triangle ACF is a right angled triangle so, we can use pythagoras theorem[tex]x^2=(\frac{x}{2} )^2+12^2[/tex]now, we can solve for x[tex]\frac{3x^2}{4}=144[/tex][tex]x^2=192[/tex][tex]x=8\sqrt{3}[/tex]Since, it is equilateral triangleso, [tex]AC=x=8\sqrt{3}[/tex]we know that E is a mid-point so, [tex]AE=\frac{AC}{2}[/tex]now, we can plug values[tex]AE=\frac{8\sqrt{3}}{2}[/tex][tex]AE=4\sqrt{3}[/tex]................Answer