Question

Triangle ABC is transformed with the center of dilation at the origin. Pre-image: △ABC with vertices A(−3, 4), B(−1, 12), C(4, −2) Image: △A'B'C' with vertices A'(−0.6, 0.8), B'(−0.2, 2.4), C'(0.8, −0.4) What is the scale factor of the dilation that maps the pre-image to the image?

Answer 1

When triangle ABC is transformed with the center of dilation at the origin. The image will become △ABC with vertices A(−3, 4), B(−1, 12), C(4, −2). The scale factor of the dilation that maps the pre-image to the image is 8:3

Answer 2

The scale factor of the dilation is 0.2 or
`(1)/(5)`.

Explanation:

The vertices of pre-image are A(−3, 4), B(−1, 12), C(4, −2).

The vertices of image are A'(−0.6, 0.8), B'(−0.2, 2.4), C'(0.8, −0.4).

If scale factor of dilation is k and center of dilation is origin, then

`P(x,y)\rightarrow P'(kx,ky)`

It is given that A(−3, 4).

`A(−3, 4)\rightarrow A'(k(-3),k(4))`

Therefore the image of A is

A'(k(-3),k(4))

A'(-3k,4k) .... (1)

It is given that the image of A is

A'(−0.6, 0.8) .... (2)

On comparing (1) and (2), we get

`-3k=-0.6`

Divide both sides by -3.

`k=(-0.6)/(-3)`

`k=0.2`

If center of dilation is origin, then the direct formula to calculate scale factor is

`k=\frac{\text{coordinate of x'}}{\text{coordinate of x}}`

`k=(-0.6)/(-3)`

`k=0.2`

Therefore the scale factor of the dilation is 0.2 or
`(1)/(5)`.