Question

**ANSWER ASAP WILL GIVE BRAINLIEST** **PICTURES SHOWN BELOW**

Answer 1

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 $|k|>1$, then the  dilation is an enlargement.  

If $|k|$, 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 = $\frac{OA'}{OA}$  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,

$\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

Then,

$k = \frac{OA'}{OA} = \frac{\sqrt{2-0)^2+(5-0)^2}}{\sqrt{(1-0)^2+(2.5-0)^2}} = \frac{\sqrt{2)^2+(5)^2}}{\sqrt{(1)^2+(2.5)^2}}$

or

$k = \frac{\sqrt{4+25}} {\sqrt{1+\frac{25}{4}}}= \frac{\sqrt{29}} {\sqrt{\frac{29}{4}}}$

Simplify:

k = $\sqrt{4} = 2$ >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:

$(x , y) \rightarrow (cx , cy)$ where c is the scale factor i.e, c =3

Or  we can write it as;

$(x , y) \rightarrow (3x , 3y)$

The images of vertices of ABCD are;

$A(3, 0) \rightarrow (3 \cdot 3 , 3\cdot 0)$ = A'(9,0)

$B(1, -2) \rightarrow (3 \cdot 1 , 3\cdot -2)$ =B'(3,-6)

$C(3, -5) \rightarrow (3 \cdot 3 , 3\cdot -5)$ = C'(9,-15)

$D(7, -1) \rightarrow (3 \cdot 7 , 3\cdot -1)$ = D'(21, -3)