Dilations

When we think of a dilation, we think of a shape expanding, as in the case of dilated pupils in your eyes. That is accurate, but can also lead to some confusion. Judy Hallock and I were trying to get the Geometer's Sketchpad to construct the dilation of a given triangle, but we were not getting the results we were seeking. Our thoughts were that if we constructed a triangle, selected it, and then dilated it, we would either obtain a larger/smaller edition of the original, or a second larger/smaller version. That was not happening. Through some more investigation, we learned that a dilation is really a copy of an original image, as we had thought, but with an additional requirement: a center point from which "rays" emanate. These rays contain corresponding points in the original and the new images. Take a look at this:

See how any ray drawn from the center (you might also think of the center as the angle's vertex) must contain corresponding points in the images? That's why in Sketchpad you must select a center point, then your shape, then dilate the shape using a scale factor - less than 1 for smaller, greater than 1 for larger.

The sketch below will allow you change the dilation by clicking and dragging point D, the center. Try bringing point D to the center of the large triangle, triangle ABC. Then, the smaller triangle will "fit inside", which is what Judy and I first expected. So going back to the dilated eyes example, they do expand, but from a center which is, obviously, the center of the pupil circle. But dilations occur even when the "center" is moved somewhere else!