Ireally need , ill

in the figure, abc is congruent to adc. if the square abcd is dilated by a factor of 2 to form a'b'c'd', what is the ratio of the area of a'b'c'd' to the area of abcd?

a) 2: 1
b) 3: 1
c) 4: 1
d) 5: 1

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)