1. Congruence and Transformations on the coordinate plane
Section 4.4 (4.1 in textbook!)

2. A transformation is a change in the position or size of a figure.
Some transformations are:
Translations (slides)
Reflections (flips)
Rotations (turns)
Dilations (changes size)

3. An Isometry or rigid transformation is a transformation that preserves length, angle measure and area. The image is congruent to the pre-image.
Translations, reflections, and rotations are all rigid transformations. The image is CONGRUENT to the pre-image.
A dilation changes the image with a scale factor so this is not a rigid transformation. The image is SIMILAR to the pre-image.

4. Dilation
If you enlarge the image, the scale factor is greater than One. If you reduce the image, the scale factor is less than one. The scale factor can’t equal 1. anyone know the reason?

6. A dilation with scale factor k > 0 and
center (0, 0) maps (x, y) to (kx, ky).
In a transformation, the original figure is the
pre-image. The resulting figure is the image.
Remember!

7. Example 1: Drawing and Identifying Transformations
Apply the transformation M to the polygon with the given vertices. Identify and describe the transformation.
A. M: (x, y) → (x - 4, y + 1)
P(1, 3), Q(1, 1), R(4, 1)
P’(1-4, 3+1)=P’(-3,4)
Q’(1-4, 1+1)=Q’(-3,2)
R’(4-4, 1+1)=R’(0,2)
Translation units left and 1 unit up
Triangle PQR is congruent to triangle P’Q’R’ because they are isometry (rigid) figures

8. Apply the transformation M to the polygon with the given vertices
Apply the transformation M to the polygon with the given vertices. Then identify and describe the transformation.
M: (x,y) (x+2, y – 5)
P(1,2), Q(4,4), R(4,2)
P’(1+2, 2-5)=P’(3,-3)
Q’(4+2, 4-5)=Q’(6,-1)
R’(4+2, 2-5)=R’(6,-3)
Translation –moved 2 units right and 5 down
Triangle PQR is congruent to triangle P’Q’R’ because they are isometry (rigid) figures

9. Example 2:
B. M: (x, y) → (x, -y)
A(1, 2), B(4, 2), C(3, 1)
A’(1,-2)
B’(4,-2)
C’(3,-1)
Reflection over x-axis
Triangle ABC is congruent to triangle A’B’C’ because they are isometry (rigid) figures

10. Apply the transformation M to the polygon with the given vertices
Apply the transformation M to the polygon with the given vertices. Then identify and describe the transformation.
M: (x,y) (-x,y)
A(1,1), B(3,2), C(3,5)
A’(-1,1)
B’(-3,2)
C’(-3,5)
Reflection over y-axis
Triangle ABC is congruent to triangle A’B’C’ because they are isometry (rigid) figures

11. M: (x,y) (-y,x) R(1,2), E(1,4), C(5,4), T(5,2) R’(-2,1) E’(-4,1)
M: (x,y) (2x,2y)
K(-1,2), L(2,2), N(1,3)
R’(-2,1)
E’(-4,1)
C’(-4,5)
T’(-2,5)
90° rotation counterclockwise with center of rotation (0, 0)
Rectangle RECT is congruent to Rectangle R’E’C’T’ because they are isometry (rigid) figures
K’(2●-1, 2●2)=K’(-2,4)
L’(2●2, 2●2)=L’(4,4)
N’(2●1, 2●3)=N’(2,6)
Dilation with scale factor 2 and center (0, 0)
Triangle KLN is similar to Triangle K’L’N’ because the image compared to the pre-image was enlarged (twice the size)

12. Example: Determining Whether Figures are Congruent
Determine whether the polygons with the given vertices are congruent. Then state the type of transformation and describe it. Give the rule.
A. A(-3, 1), B(2, 3), C(1, 1) vs.
P(-4, -2), Q(1, 0), R(0, -2)
The triangle are congruent; △ ABC can be mapped to △PQR by a translation: (x, y) → (x - 1, y - 3).

13. Example: Continued
B. A(2, -2), B(4, -2), C(4, -4) vs.
P(3, -3), Q(6, -3), R(6, -6).
The triangles are not congruent; △ ABC can be mapped to △ PQR by a dilation with scale factor k ≠ 1: k=1.5
(x, y) → (1.5x, 1.5y)
enlargement

14. Determine whether the figures are congruent.
Then state the type of transformation and describe it.
Give the rule.
A(1,1), B(4,1), C(4,3) vs.
P(-4,2), Q(-1,2), R(-1,4)
A(2,2), B(-4,4), C(2,4) vs.
P(3,3), Q(-6,6), R(3,6)
The triangle are congruent; △ ABC can be mapped to △PQR by a translation: (x, y) → (x - 5, y + 1).
The triangles are not congruent; △ ABC can be mapped to △ PQR by a dilation with scale factor k ≠ 1: k=1.5
(x, y) → (1.5x, 1.5y)
enlargement

15. 3. A(2,-1), B(3,0), C(2,3) vs.
P(1,2), Q(0,3), R(-3,2)
4. A(-2,-2), B(-4,-1), C(-1,-1) vs.
T(2,2), U(4,1), V(1,1)
The triangle are congruent; △ ABC can be mapped to △PQR by a 90° rotation counterclockwise about the origin: (x, y) → (-y, x).
The triangle are congruent; △ ABC can be mapped to △PQR by a 180° rotation about the origin:
(x, y) → (-x, -y).

16. 5. A(-4,4), B(-4,6), C(2,6), D(2,4) vs.
W(-2,2), X(-2,3), Y(1,3), Z(1,2)
The triangles are not congruent; Rectangle ABCD can be mapped to Rectangle WXYZ by a dilation with scale factor k ≠ 1: k=0.5
(x, y) → (0.5x, 0.5y)
reduction