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

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)

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.

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?

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!

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 -- 4 units left and 1 unit up Triangle PQR is congruent to triangle P’Q’R’ because they are isometry (rigid) figures

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

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

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. 2. 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

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)

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).

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

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

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).

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