Since the scale factor is #2#, this means all the lengths in the bigger triangle are twice as long as they are in the smaller triangle.
In the case of #\DeltaADE# and #\DeltaABC#, altitude #\bar{CG}# is twice as long as altitude #\bar{EF}# because of the scale factor of #2#.
To see what this means for the area, we can solve it algebraically.
Say the base and height of #DeltaABC# (the bigger triangle) are #x# and #y#. Now we know that the base and height of #DeltaADE# are #1/2x# and #1/2y# because of the reasons described before.
We also know that the formula for the area for the area of a triangle is #1/2bh#.
We can write a system of equations now:
#{ (A_{DeltaABC} =1/2xy, qquad(1)), (A_{DeltaADE} = 1/2(1/2x)(1/2y)=1/8xy, qquad(2)) :}#
To compare equations #(1)# and #(2)#, we can write two ratios:
#A_{DeltaABC}:A_{DeltaADE}=1/2xy:1/8xy#
#(A_{DeltaABC})/(A_{DeltaADE})=(1/2xy)/(1/8xy)#
#(A_{DeltaABC})/(A_{DeltaADE})=(1/2color(red)(cancel(color(black)(xy))))/(1/8color(red)(cancel(color(black)(xy))))#
#(A_{DeltaABC})/(A_{DeltaADE})=(1/2)/(1/8)#
#(A_{DeltaABC})/(A_{DeltaADE})=1/2*8/1#
#(A_{DeltaABC})/(A_{DeltaADE})=8/2=4#
This means that #DeltaABC#'s area is actually #4# times as much as #DeltaADE#'s area.
