Since similar triangles have the same combination of angles, they bear the same shape. One triangle essentially is a bigger version of the other.
The ratio of the two bases, like that of the two heights, is the same as the ratio of each pair of corresponding sides.
Given 2 similar triangles where each side of the larger triangle is x times the corresponding side of the smaller triangle:
If B = base of the larger triangle and b = base of the smaller triangle, B = xb.
If H = height of the larger triangle and h = height of the smaller triangle, H = xh.
Since the area of the larger triangle = (1/2)BH = (1/2)(xb)(xh) = x²(1/2)bh, the area of the larger triangle is x² times the area of the smaller triangle.
In the problem above, since DA:DC = 3:1, B = 3b and H = 3h, implying that the area of ADG is 3²=9 times the area of CDE.
Since CDE = 42, ADG = 9*42 = 378.
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