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 [tex]|k|>1[/tex], then the dilation is an enlargement.
If [tex]|k|<11[/tex], 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 = [tex]\frac{OA'}{OA}[/tex] 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,
[tex]\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]
Then,
[tex]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}}[/tex]
or
[tex]k = \frac{\sqrt{4+25}} {\sqrt{1+\frac{25}{4}}}= \frac{\sqrt{29}} {\sqrt{\frac{29}{4}}}[/tex]
Simplify:
k = [tex]\sqrt{4} = 2[/tex] >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:
[tex](x , y) \rightarrow (cx , cy)[/tex] where c is the scale factor i.e, c =3
Or we can write it as;
[tex](x , y) \rightarrow (3x , 3y)[/tex]
The images of vertices of ABCD are;
[tex]A(3, 0) \rightarrow (3 \cdot 3 , 3\cdot 0)[/tex] = A'(9,0)
[tex]B(1, -2) \rightarrow (3 \cdot 1 , 3\cdot -2)[/tex] =B'(3,-6)
[tex]C(3, -5) \rightarrow (3 \cdot 3 , 3\cdot -5)[/tex] = C'(9,-15)
[tex]D(7, -1) \rightarrow (3 \cdot 7 , 3\cdot -1)[/tex] = D'(21, -3)
