Exercise 9 Page 59

Let's take a look at the given diagram.

We are told that the red piece is rotated about the origin and then translated. After these transformations, it moves to the place marked by the dashed lines. We want to find the coordinates of the vertices of this piece after the rotation. We need to follow two steps to find the coordinates.

1. Identify the rotation.
2. Find the coordinates of the vertices after the rotation.

   Let's do it!

   Identifying the Rotation

   Let's look at the given diagram again. Remember that our goal is to identify the rotation that turns the red piece into a position so that it can be translated into the place marked by the dashed lines.
   
   When we rotate the piece clockwise by 90^(∘) about the origin it lines the piece up so that it can be translated down to the dashed space. Then the piece can be translated down by 6 units to get to its final placement.

   Finding the Coordinates

   Next, let's focus on the red piece after the rotation so that we can find the coordinates of the vertices. We can mark its vertices and their coordinates.

   

   Finally, let's make a list of these coordinates. (1,-1), (0,-1), (0,0), (-1, 0), (-1,1), (0,1), (0,2), (1,2) Note that this is one of many ways that we could write our answer. We could change the order of the points and the list would still be correct.

   Alternative Solution

   Using the Coordinates of the Original Figure

   We can also find the coordinates of the vertices in another way. Just like before, we will first find the rotation that allows the piece to fall into the correct place.

   

   Let's mark and label the vertices of the original red piece.

   

   The vertices of the original red piece have the following coordinates. (0,0), (0,-1), (-1, -1), (-1,0), (-2,0), (-2,1), (1,1), (1,0) We know that the first transformation is a clockwise rotation by 90^(∘) about the origin. During this type of rotation, the coordinates of points always change in the same way. ( x, y) ⟶ ( y, - x) Let's change the coordinates of the original piece using this algebraic rule. This will give us the coordinates of our shape after the rotation.

   (x,y)	(y, - x)	Simplify
   ( 0, 0)	( 0, - 0)	(0,0)
   ( 0, -1)	( -1, - 0)	(-1,0)
   ( -1, -1)	( -1, -( -1))	(-1,1)
   ( -1, 0)	( 0, -( 1))	(0,1)
   ( -2, 0)	( 0, -( -2))	(0,2)
   ( -2, 1)	( 1, -( -2))	(1,2)
   ( 1, 1)	( 1, - 1)	(1,-1)
   ( 1, 0)	( 0, - 1)	(0,-1)

   Finally, let's list the points. (0,0), (-1,0), (-1,1), (0,1), (0,2), (1,2), (1,-1), (0,-1) We found the same set of points, although they are written in a different order, that we found using the first method!