If triangle CDE is dilated by a scale factor of 1/2 with a center of dilation at the origin, how does the area of C'D'E' compare with the area of CDE?
a
The area of C'D'E' is 1/2 times the area of CDE.
b
The area of C'D'E' is 1/4 times the area of CDE.
c
The area of C'D'E' is 1/8 times the area of CDE.
d
The area of C'D'E' is 1/16 times the area of CDE.
b
The area of C'D'E' is 1/4 times the area of CDE.
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To determine the relationship between the areas of the two triangles, we need to understand how the area of a triangle changes under dilation.
When a triangle is dilated by a scale factor of k, the area of the dilated triangle is given by the square of the scale factor (k^2) multiplied by the area of the original triangle.
In this case, the scale factor is 1/2, so the area of the dilated triangle C'D'E' will be (1/2)^2 times the area of the original triangle CDE.
Simplifying this expression, we get:
(1/2)^2 = 1/4
Therefore, the area of triangle C'D'E' is 1/4 times the area of triangle CDE.
So, the correct answer is option b: The area of C'D'E' is 1/4 times the area of CDE.