Homotecia

	HOMOTHETY (ENLARGEMENT & REDUCTION)
	Geometry

1. WHAT IS HOMOTHETY?

Homothety or enlargement, of centre O and ratio or scale factor (which is not 0) k, is the transformation which associates point A with A' ,which is in line with A and O, such that: OA´=k·OA. When k>0 the transformation is called direct homothety and when k<0 the transformation is called inverse homothety.

1.- Move the centre of enlargement O with the mouse and watch how its function changes in the homothetic triangle (or image).

2.- Change the scale factor making it bigger and smaller. What happens when k=1? What about when k=-1?

2. THE CENTRE OF ENLARGEMENT AT THE ORIGIN OF THE AXES.

When the centre of enlargement is at the origin it is easy to find the relation between the coordinates of points and their homothetic images. The relation between point A(x,y) and its image A´(x´,y´) is: x´=kx y´=ky

Check this relation in the Descartes window by altering the coordinates for points A and B. The coordinates of the image are given in the top left-hand corner of the window.

3.- In your exercise book draw a triangle with points A(1,2), B(3,2) and C(-1,3) and its image, using the origin as the centre of enlargement and scale factor k=2 and carrying out the transformation on each of its coordinates.

3. combining TWO homothetIC TRANSFORMATIONS with the same centre
When two homothetic transformations with the same centre of enlargement are combined the result is one transformation whose ratio or scale factor is the product of the other two ratios. In the Descartes window below you can see how the combination of transformations with ratios k1 and k2, illustrated in the left-hand window, is equal to the transformation with ratio k=k1k2, illustrated in the right-hand window.

4.- Work out the combination of homothetic transformations with ratios k1=0.7 and k2=1.8. Use the right-hand window to check that your results are correct. Repeat the exercise for another set of ratios: k1=-0.25 and k2=-3.

	Miguel García Reyes

	Spanish Ministry of Education. Year 2001

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