We can see that the pre-image has side lengths of 32 and 16, and the image has side lengths of 4 and 2. This indicates that the pre-image has been reduced. To find the scale factor we can divide one of the lengths of the image by the corresponding side length of the pre-image.

\dfrac{C'D'}{CD}=\dfrac{4}{32} Simplifying the quotient gives a scale factor of \dfrac{1}{8}.

Since the center of dilation is the origin, (0,0), we can take the coordinate of each point in the pre-image and multiply them by the scale factor to get the vertices of the image.

This means that:

(4, 8) \to (16, 32)

(10, 8) \to (40, 32)

(4, 14) \to (16, 56)

This transformation can be represented by the function notation D_{4,O} or by the coordinate form \left(x,y\right)\to\left(4x,4y\right).

One way to approach this problem is by sketching an example so we can see how an object will change based on the transformation done to it. Let's use a 2 \times 2 square on the coordinate plane as an example.

We can see that distance has not been preserved since the side lengths of the image are larger than the preimage, however the angle measurements appear to not have changed. Therefore a translation to the right 3 units, followed by a dilation using a scale factor of 2 preserves only angle measurements.

We can be more confident that the angle measures have not changed if we do a construction using technology.

First we can construct the image after the translation, but before the dilation using the Polygon tool.

Then we can measure all of the angles in this polygon by selecting the Angle tool and clicking somewhere in the middle of the polygon.

Next, we can using the Dilate from Point tool and click on the polygon, then the point A and then type in 2 to perform the required dilation.

Finally, we can use the Angle tool to find all of the angle measures on the final image.

We can then explore this construction by moving around the points on ABCD to see what happens to A'B'C'D'.