Question

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?

Answer 1

Answer: The area of C'D'E' = 1/4 × the area of CDE

Step-by-step explanation:

Let the coordinates of triangle CDE are $(x_1,y_1)$, $(x_2, y_2)$ and $(x_3,y_3)$

Since, In the dilation about origin by the scale factor k,

$(x,y) \rightarrow (kx,ky)$

Thus, when triangle CDE is dilated by a scale factor $\frac{1}{k}$

Then the coordinates of triangle C'D'E' are,

$(\frac{x_1}{2},\frac{y_1}{2})$,$(\frac{x_2}{2},\frac{y_2}{2})$ and $(\frac{x_3}{2},\frac{y_3}{2})$

Since, the area of triangle C'D'E'

= $\frac{1}{2} [\frac{x_1}{2} (\frac{y_2}{2} - \frac{y_3}{2}) + \frac{x_2}{2} (\frac{y_3}{2} - \frac{y_1}{2})+\frac{x_3}{2} (\frac{y_1}{2} - \frac{y_2}{2})]$

=  $\frac{1}{4}[\frac{1}{2}(x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)]$

= $\frac{1}{4} \times \text{ area of triangle CDE}$