**Two ways to rotate a triangle
90' CW**

**Alvaro, a 7th grader, was working on
transformations of a cap in his textbook. He was changing the x- and y-coordinates of the
original shape to get the new shape. Then he made a triangle 1/5 the coordinates
of the original triangle whose vertices were (0,5), (5,8), and (10,2)- see
below. The resulting small triangle in the diagram had vertices (0,1), (1,1.6)
and (2,0.4). Notice the small triangle's vertices were each on a line from the
original vertices to the origin.**

**Then Don asked
Alvaro to rotate the
original triangle 90****'
clockwise. Don worked with Alvaro using a compass to draw arcs with radii from
the origin to each vertex. Then using a protractor they made 90' angles and drew
lines to intersect the arcs to get the new vertices. These resultant vertices of
the rotated triangle were very close to (5,0), (8, ^{-
}5) and (2,^{- }10).**

**While Don and Alvaro did
this, Ian, a 6th grader, was listening and proceeded to do the rotation with a
matrix below:**

**Ian's matrix acts
like the complex number **^{
- }**i.
Ian has been working on the problem of transforming a
dog, using the 81-2x2 matrices (formed with only 0's, 1's and **
^{-}1's as discussed in
Don's book "Changing Shapes With Matrices").

**Great job
guys!!**