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

Answer:

The scale factor of the dilation is 0.2 or $\frac{1}{5}$.

Step-by-step 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.

$k=\frac{-0.6}{-3}$

$k=0.2$

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

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

$k=\frac{-0.6}{-3}$

$k=0.2$

Therefore the scale factor of the dilation is 0.2 or $\frac{1}{5}$.

Answer 2

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