1. 9.4 Properties of Rotations
POD:
2 3 5 −7 −1 −2 3 −4
−7 + −1 −2 3 −4
−7 − −1 −2 3 −4

2. Obj: Rotate figures about a point
Rotation – a transformation is which a figure is turned about a fixed point.
Center of rotation – the fixed point in which a figure is turned about.
Angle of rotation – formed from rays drawn from the center of rotation to a point and its image

3. If P is the center of
rotation then mQPQ’
= the rotation.
If Q is the center of
rotation, then the
image of Q = Q

4. Graph and draw a rotation
1. Draw a 100 rotation of ABC about P.

6. Coordinate Rules :
When point (a,b) is rotated ccw about the origin…
1. rotation of 90
(a,b) => (-b,a)
2. rotation of 180
(a,b) => (-a, -b)
3. rotation of 270
(a,b) => (b, -a)

7. Graph using coordinates
Graph quadrilateral RSTU with vertices R(1,-1), S(3,-1), T(3,-5) and U(0,-4). Then rotate the quadrilateral 270 about the origin.
Graph JKL with vertices J(4,1), K(5,4) and L(7,0). Rotate the triangle 90 about the origin.

8. Using Matrices Rotation Matrices CCW 90 rotation 180 rotation
0 − −1 0 0 −1
270 rotation  rotation
0 1 −

9. Trapezoid EFGH has vertices E(-4, 1), F(-4, 3), G(0, 3), and H(1, 1)
Trapezoid EFGH has vertices E(-4, 1), F(-4, 3), G(0, 3), and H(1, 1). Find the image matrix for a 180° rotation of EFGH about the origin. Graph EFGH and its image.

10. Theorem…
Theorem 9.3 Rotation Theorem – A rotation is an isometry

11. Cases of Theorem 9.3
Page 593
Need to prove the rotation preserves the length of a segment. Consider a segment QR rotated about point P to produce segment Q’R’.
Case 1 – R, Q and P are noncollinear.
Case 2 – R, Q, and P are collinear.
Case 3 – P and R are the same point

12. Prove it…
Find the value of y in the rotation of the quadrilateral.

13. Find the value of y in the rotation of the quadrilateral.
By Theorem 9.3, the rotation is an isometry, so corresponding side lengths are equal. Then 3x = 9, so x = 3. Now set up an equation to solve for y.
4y = 6x-2
4y = 6(3) -2
y = 4