Mathematical Enlargements (Dilations)
Learn about mathematical enlargements (mathematical dilations), reductions, projections and inversions in this interactive Java(TM) applet. Learn also about centres of enlargement (centers of dilation), scale factors (ratios), positive enlargements (positive dilations), negative enlargements (negative dilations) and mathematical similarity.
Author and programmer: Ron Barrow
UK Years 7-11, KS3, KS4, Foundation and Higher GCSE Mathematics - Geometry, Shape and Space
US Grade 9 - Geometry
How to Use this Applet
On the screen you will see a blue shape and a green one. The green one is a mathematical enlargement or mathematical dilation of the blue one. At the beginning the enlargement (dilation) has scale factor (ratio) of 2 and centre (0,0). This program allows to change the shape, the scale factor and the centre (center), so that you can investigate mathematical enlargements (dilations). You can move the blue shape (the object) around the screen, by clicking the red dots at its vertices (corners) and dragging them with the mouse. The green shape (the image) will also change. You can also see the effects on the co-ordinates of the vertices at the top left of the screen. Checking the "center" box will show or hide a dot which marks the centre (center). If you can see this dot you can click-and-drag it and so move the center. When you do so you will see the green image move while the blue object stays still. Try it. The dots on the image vertices can be removed by unchecking the "vertex dots" box. If you check the "integers" box then the image, the centre and the scale factor will have integer (whole number) values only. This might help you see what's going on. Checking the "rays" box will show or hide the grey lines (rays) which connect all the vertices to the center, checking the "image" box will show or hide the green image shape, and checking the "scale factor (ratio)" box will show or hide the scale factor (ratio) information, checking the "grid" box will show or hide the co-ordinate gridlines. The "axes" button hides or shows the axes. You can change the scale factor (ratio), and watch the effect on the screen. Notice what happens when the scale factor is negative. You can also change the scale of the whole screen with the "zoom" slider by dragging it with your mouse. Pressing the "reset" button returns everything to the start position. You should know that the two shapes, the image and the object are always mathematically similar. The ratios of the sides are constant and the angles made at the vertices are unchanged. Play around! You will soon get a feel of how mathematical enlargements (dilations) behave by watching closely. There is a great deal to learn from this applet, so take your time.