When the lengths (1-dimensional) change by a factor 1.5 (the definition of dilating a figure), the area (2-dimensional) will change by a factor 1.5 to the power of the quotient of the values of the dimensions, which is 2/1 = 2. So the area will change by a factor of 1.5^2 = 2.25.
In this question, the square is dilated by a factor of 2 to form A'B'C'D'. That mean the new square sides will be twice as much as the old square. The formula for area is side^2. Then, the area of the new square should be
a'=2a
old square area= a*a= a^2 new square area= a'*a'= 2a*2a= 4a^2
Well, first of all, the first statement (ABC = ADC) looks like it just says that the two halves of the little square ... each side of the diagonal ... are congruent. That's no big deal, and it's no help in answering the question.
The effect of the dilation is that all the DIMENSIONS of the square are doubled ... each side of the square becomes twice as long.
Then, when you multiply (length x width) to get the area, you'd have
Area = (2 x original length) x (2 x original width)
and that's the same as (2 x 2) x (original length x original width)
= (4) x (original area) .
Here's an easy, useful factoid to memorize:
-- Dilate a line (1 dimension) by 'x' times . . . multiply the length by x¹
-- Dilate a shape (2 dimensions) by 'x' . . . multiply area by x²
-- Dilate a solid (3 dimensions) by 'x' . . . multiply volume by x³
And that's all the dimensions we have in our world.
Oh, BTW . . .
-- Dilate a point (0 dimensions) by 'x' . . . multiply it by x⁰ (1)
In this question, the square is dilated by a factor of 2 to form A'B'C'D'. That mean the new square sides will be twice as much as the old square. The formula for area is side^2. Then, the area of the new square should be
a'=2a
old square area= a*a= a^2 new square area= a'*a'= 2a*2a= 4a^2
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