Teacher Notes    

CONCEPT 1 – ALL CIRCLES ARE SIMILAR

 

 

In an earlier objective we defined two figures to be similar if there was a single or sequence of similar transformations that could map the one figure onto the other.  The similarity transformations consist of translations, rotations, reflections and dilations.  These all preserve the shape of the figure.  Due to unique nature of a circle – that it is defined by a single length, the radius, it seems quite obvious that we could use a scale factor to alter the radius in such a way that it could be the same size as another circle.
Teacher Note -- Nothing surprising here... just a nice review of similarity and the similarity transformations. They could actual get tucked into Unit #2 if you wanted to.

 

A sequence of similarity transformation that would always work to establish two circles to be similar would be a translation vector from one center to the other.  This would form two concentric circles (circles with the same center).  Once the circles share a common center then from that center point we could perform a dilation of R:r or r:R… both of these scale factors would map one circle onto the other.

 

Using the scale factor R:r would be a reduction and would map the large circle onto the smaller circle.  Using the scale factor r:R would be an enlargement and would map the small circle onto the large circle.

 

Either of these establish that using only similarity transformations, a translation and a dilation, we could map any circle onto another….

 

THUS ALL CIRCLES ARE SIMILAR.

 

If we use a coordinate plane we can be more specific as two our translation and our scale factor.

 

Thus using two similarity transformations, a translation of <6,-2> and a dilation about B by a scale factor of 3/2  we are able to map circle A onto circle B.