Answer:
3.
Dilation is a transformation that produces an image that is of same shape as the original, but in a different size.
*The value of scale factor ( i.,e k) determines whether the dilation is an enlargement or a reduction.
If
, then the dilation is an enlargement.
If
, then the dilation is a reduction.
To find the scale factor:
In other words, a dilation is a rule that moves points in the plane a specific distance, determined by the scale factor k, from a center O.
Labelled the Blue triangle as A',B', and C'
Also, for black triangle as A, B, and C as you can see the figure shown below in the attachment.
Scale factor is given by;
k =
where O is the origin.
From the figure, A =(1, 2.5) and A' = (2,5)
By distance formula we can find the value of OA' and OA i.e,
![\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}](https://tex.z-dn.net/?f=%5Csqrt%7B%28x_2-x_1%29%5E2%2B%28y_2-y_1%29%5E2%7D)
Then,
![k = \frac{OA'}{OA} = \frac{\sqrt{2-0)^2+(5-0)^2}}{\sqrt{(1-0)^2+(2.5-0)^2}} = \frac{\sqrt{2)^2+(5)^2}}{\sqrt{(1)^2+(2.5)^2}}](https://tex.z-dn.net/?f=k%20%3D%20%5Cfrac%7BOA%27%7D%7BOA%7D%20%3D%20%5Cfrac%7B%5Csqrt%7B2-0%29%5E2%2B%285-0%29%5E2%7D%7D%7B%5Csqrt%7B%281-0%29%5E2%2B%282.5-0%29%5E2%7D%7D%20%3D%20%5Cfrac%7B%5Csqrt%7B2%29%5E2%2B%285%29%5E2%7D%7D%7B%5Csqrt%7B%281%29%5E2%2B%282.5%29%5E2%7D%7D)
or
![k = \frac{\sqrt{4+25}} {\sqrt{1+\frac{25}{4}}}= \frac{\sqrt{29}} {\sqrt{\frac{29}{4}}}](https://tex.z-dn.net/?f=k%20%3D%20%5Cfrac%7B%5Csqrt%7B4%2B25%7D%7D%20%7B%5Csqrt%7B1%2B%5Cfrac%7B25%7D%7B4%7D%7D%7D%3D%20%5Cfrac%7B%5Csqrt%7B29%7D%7D%20%7B%5Csqrt%7B%5Cfrac%7B29%7D%7B4%7D%7D%7D)
Simplify:
k =
>1
therefore, by definition of scale factor, the dilation is an enlargement.
4.
Given: the center of dilation at origin (0,0) with scale factor (c) = 3.
The vertices of ABCD are;
A = (3,0)
B = (1, -2)
C = (3 , -5) and
D = (7 , -1)
The rule of dilation with center of dilation is given by:
where c is the scale factor i.e, c =3
Or we can write it as;
The images of vertices of ABCD are;
= A'(9,0)
=B'(3,-6)
= C'(9,-15)
= D'(21, -3)