# An Investigation ofthe Nine Point Circle

Click on the sketch below to explore the meaning of dilation.

Dilating an object makes it bigger or smaller without changing it's shape. In order to do a dilation, you need three pieces of information...

• The center of dilation.
• The object to be dilated.
• The dilation ratio (how much to shrink or expand the object.

Click on the sketch below to learn to dilate an object using Sketchpad. The instructions are below the picture.

In this sketch you have been provided with an object to dilate (the big arrow) and a point C to use as a center of dilation. You have also been given a point P on the object so you can do some mathematical exploration of the dilation process. Use the following procedures...

• Select the center of dilation (Point C) and mark it by pulling down the Transform Menu to Mark Center. Point C will respond by expanding and contracting.
• Select the arrow and point P. Pull down the Transform Menu to Dilate. You will get a box that allows you to type in the dilation ratio. Type in New: 1, Old: 4, and return to perform the dilation.

Now investigate the dilation process...

• Draw and measure segments CP' and CP.
• Calculate the ratio CP'/CP. How does this ratio relate to the 1 and 4 you entered? (Check your answer.)
• Move the center C. Does the ratio change? (Check your answer.)
• Redo the dilation a ratio of 2 (new) to 1 (old). How is this different from your first dilation. (Check your answer.)

There is some mathematical terminology about dilations that you should learn because we will use it again. Your original object before dilation is called a pre-image and the object that results from the dilation is the image. The images of points like P are often designated as P' for the first dilation, P'' for the second, etc.

After this investigation, you should know how to dilate an object using Sketchpad. You should also have some idea how the dilation ratio is related to the result of the dilation. If you need to experiment a little more, please feel free to play with this sketch and try some other dilations before you return to the Nine Point Circle.

Next: Go Back