a Let's begin by drawing the described polygon ABCD.
The rotation of a polygon 90^(∘) clockwise about the origin changes the coordinates of the figure's vertices in the following way.
preimage (a,b)→ image (b,- a)
Using this rule on ABCD, we can find the vertices of A'B'C'D',
Point
(a,b)
(b,- a)
A
(0,3)
(3,0)
B
(2,5)
(5,-2)
C
(6,3)
(3,-6)
D
(4,1)
(1,-4)
Now we can draw A'B'C'D',
Examining the diagram, we see that B' has the coordinates (5,- 2).
b Let's translate A'B'C'D' by 8 units up and 7 units left.
Examining the diagram, we see that C'' has the coordinates (- 4,2).
c If something maps onto itself after a 180^(∘) rotation, it means it has a 180^(∘) rotational symmetry. To do that, we have to rotate it around its center. Since this is a parallelogram, the center is the point of intersection of the diagonals.
Examining the diagram, we see that the point of rotation has the coordinates (3,3).
