Transformations on triangles?

Question

The figure (http://img442.imageshack.us/img442/1415/chapter5testnumber2ys9.png) shows a triangle in the coordinate plane along with a smaller triangle formed by joining the midpoints of the sides of the larger triangle. How could you use a series of transformations of triangle ABC to produce triangle DEF? Write a rule (in the form P(x,y)--->P'(?,?) ) for each transformation in your series that shows how the coordinates of the transformed figure are obtained. Use your rules to find the coordinates of each vertex of the triangle after each transformation. Lastly, write a rule that would transform ABC into DEF by just one transformation.

Answer 1

I can't get your picture but here is what I think it is:
Draw a triangle ABC. Let D be the midpoint of AB, E be the midpoint of BC, and F be the midpoint of AC. Now draw DE, Ef and FD forming the triangle DEF.

Triangle DEF is exactly 1/4 the size of triangle ABC. This is because triangles FCE, AFD, DEB, and EDF are all congruent.

The easiest way to transform ABC to DEF is to draw ED and then say triangle BDE is the required triangle. It is, but it is not in the same orientation as ABC. If you insist it be in the same orientation, then rotate BDE 180 degrees clockwise about the point E.

Anther way would be to reflect ABC across AB and then dilate the result using a dilation factor of 1/2.

Assuming A is at the origin then the transformation could be made by C(x,y) --> C'(x/2,y/2) and B(x1,y1)--> B'(x1/2,y1/2) and A stays where it is.