Question

Select the correct answer.What is the image of this figure after this sequence of dilations?1. dilation by a factor of -1 centered at the origin2. dilation by a factor of 2 centered at (-1,1)

Answer 1

The coordinates of the original figure are:

(-2,1)

(3,1)

(1,3)

(-2,3)

A dilation by a negative scale factor produces an image on the other side of the center of enlargement.

As the first dilation is by a factor of -1 centered at the origin, the length of the sides doesn't change, but the new coordinates will be:

$(x,y)\to(kx,ky)$

Apply this to the given coordinates:

$\begin{gathered} (-2,1)\to(-1\cdot-2,-1\cdot1)\to(2,-1) \\ (3,1)\to(-1\cdot3,-1\cdot1)\to(-3,-1) \\ (1,3)\to(-1\cdot1,-1\cdot3)\to(-1,-3) \\ (-2,3)\to(-1\cdot-2,-1\cdot3)\to(2,-3) \end{gathered}$

The image after the first dilation looks like this:

Now, the second dilation is by a scale factor of 2, centered at (-1,1).

As it is not centered in the origin, we can use the following formula:

$(x,y)\to(k(x-a)+a,k(y-b)+b)$

Where k is the scale factor and (a,b) are the coordinates of the center of dilation.

By applying this formula to the actual coordinates we obtain:

$\begin{gathered} (2,-1)\to(2(2-(-1))+(-1),2(-1-1)+1)\to(5,-3) \\ (-3,-1)\to(2(-3-(-1))+(-1),2(-1-1)+1)\to(-5,-3) \\ (-1,-3)\to(2(-1-(-1))+(-1),2(-3-1)+1)\to(-1,-7) \\ (2,-3)\to(2(2-(-1))+(-1),2(-3-1)+1)\to(5,-7) \end{gathered}$

If we place these coordinates in the coordinate plane we obtain:

The answer is option B.