Transformations Chapter 4

Translations I can translate a figure in a coordinate plane.

Translations Vocabulary (page 94 in Student Journal) vector: a quantity that has both direction and magnitude (size), and is represented by an arrow drawn from 1 point to another initial point: starting point of the vector terminal point:

Translations terminal point: ending point of the vector vertical component: up and down horizontal component: left to right component form: combines the horizontal and vertical component

Translations transformation: a change in the position, shape, or size of a geometric figure image: the resulting figure after the transformation preimage: the original figure

Translations translation: sliding all points of a figure the same distance and direction rigid motion (isometry): a transformation that preserves distance and angle measures composition of transformations: a combination of 2 or more transformations

Translations Examples (space on page 95 in Student Journal) a) Name the vector in the diagram and name its component.

Translations Solutions a) Vector PQ, <-4, 5>

Translations b) Translate triangle ABC using the vector <-1, -2> if the vertices of the triangle are A(0, 3), B(2, 4) and C(1, 0).

Translations Solution b) A’(-1, 1), B’(1, 2), C’(0, -2)

Translations c) Write a rule for the translation of triangle ABC to the triangle A’B’C’ based on the graph.

Translations Solution c) (x, y)  (x + 2, y – 1)

Translations d) Graph segment RS with endpoint R(-8, 5) and S(-6, 8). Graph its image after the composition of translations below. (x, y)  (x – 1, y + 4) (x, y)  (x + 4, y – 6)

Translations Solution d)

Reflections I can reflect a figure in a coordinate plane.

Reflections Vocabulary (page 99 in Student Journal) reflection: a transformation that uses a line line a mirror to reflect a figure line of reflection: the mirror type line

Reflections glide reflection: a transformation involving a translation followed by a reflection line symmetry: a plane figure that can be mapped onto itself by a reflection in a line line of symmetry: the line of reflection

Reflections Core Concepts (page 100 in Student Journal) Coordinate Rules for Reflections Preimage Line of reflection Image (a, b) x-axis (a, -b) y-axis (-a, b) y = x (b, a) y = -x (-b, -a)

Reflections Reflection Postulate (Postulate 4.2) A reflection is a rigid motion.

Reflections Examples (space on pages 99 and 100 in Student Journal) Graph triangle ABC with vertices A(1, 3), B(5, 2), C(2, 1) and its image after the described reflection. in the line x = -1 in the line y = 3

Reflections Solutions a) b)

Reflections Graph segment AB with endpoints A(3, -1) and B(3, 2) and its image after the described reflection. c) in the line y = x d) in the line y = -x

Reflections Solutions c) d)

Reflections e) Graph triangle ABC with vertices A(3, 2), B(6, 3) and C(7, 1) and its image after the glide reflection below. (x, y)  (x, y – 6), reflection in the y-axis

Reflections Solution e)

Reflections f) How many lines of symmetry does the triangle in the diagram have?

Reflections Solution f) 3

Rotations I can rotate a figure in a coordinate plane.

Rotations Vocabulary (page 104 in Student Journal) rotation: a transformation in which a figure is turned about a fixed point center of rotation: the fixed point the figure is rotated about

Rotations angle of rotation: the angle formed by rays drawn from the center of rotation to a point and its image rotational symmetry: a plane figure that can be mapped onto itself by a rotation of 180 degrees or less about its center center of symmetry:

Rotations center of symmetry: the center of the figure

Rotations Core Concepts (page 105 in Student Journal) Coordinate Rules for Rotations about the Origin Preimage Angle of Rotation Image (a, b) 90 degrees counterclockwise (-b, a) 180 degrees counterclockwise (-a, -b) 270 degrees counterclockwise (b, -a)

Rotations Rotation Postulate (Postulate 4.3) A rotation is a rigid motion.

Rotations Examples (space on pages 104 and 105 in Student Journal) a) Draw a 60 degree rotation for triangle ABC about point P.

Rotations Solution a)

Rotations b) Graph triangle ABC with vertices A(3, 1), B(3, 4) and C(1, 1) and its image after a 180 degree rotation about the origin.

Rotations Solution b)

Rotations c) Graph segment RS with endpoints R(1, -3) and S(2, -6) and its image after the composition below. 180 degree rotation about the origin and a reflection in the y-axis

Rotations Solution c)

Rotations Does the figure have rotational symmetry? If so, describe the rotation that maps the figure onto itself. d) e) f)

Rotations Solutions d) yes, 120 degrees e) no f) yes, 180 degrees

Congruence and Transformations I can identify congruent figures and describe congruence transformations.

Congruence and Transformations Vocabulary (page 109 in Student Journal) congruent figures: figures that have the same shape and size congruence transformation: another name for a rigid motion because the preimage and the image are congruent

Congruence and Transformations Core Concepts (page 109 in Student Journal) Reflections in Parallel Lines Theorem (Theorem 4.2) If lines k and m are parallel, then a reflection in line k followed by a reflection in line m is the same as a translation.

Congruence and Transformations Reflections in Intersecting Lines (Theorem 4.3) If lines k and m intersect at point P, then a reflection in line k followed by a reflection in line m is the same as a rotation about point P.

Congruence and Transformations Example (space on page 109 in Student Journal) a) Identify any congruent figures in the diagram.

Congruence and Transformations Solution a) Triangle ABC is congruent to triangle PQR

Congruence and Transformations b) Describe a congruence transformation that maps ABCD to PQRS.

Congruence and Transformations Solution b) 90 degree rotation about the origin

Congruence and Transformations In the diagram, PJ = 3 and LP’’ = 8. c) Name any segments congruent to each other. d) Find PP’’.

Congruence and Transformations Solutions c) segments PQ, P’Q’ and P’’Q’’, segments PJ and P’J, segments P’L and P’’L, segments QK and Q’K, segments Q’M and Q’’M, segments JK and LM, segments JL and KM d) 22 units

Dilations I can identify and perform dilations.

Dilations Vocabulary (page 114 in Student Journal) dilation: a transformation in which a figure is enlarged or reduced with respect to a fixed point center of dilation: the fixed point for an enlargement or reduction

Dilations scale factor: the ratio of the lengths of the corresponding sides of the image and the preimage enlargement: occurs when the scale factor is greater than 1 reduction: occurs when the scale factor is less than 1

Dilations reduction: occurs when the scale factor is less than 1

Dilations Core Concepts (page 115 in Student Journal) Coordinate Rules for Dilations If P(x, y) is the preimage of a point, then its image after a dilation centered at the origin (0,0) with scale factor k is the point P’(kx, ky).

Dilations Examples (space on pages 114 and 115 in Student Journal) Find the scale factor. Then determine is the dilation is an enlargement or reduction. a) b)

Dilations Solutions 1/3, reduction 5/2, enlargement

Dilations c) Graph triangle PQR with vertices P(0, 2), Q(1, 0) and R(2, 2) and its image after a dilation with a scale factor of 3.

Dilations Solution c)

Dilations d) Graph triangle PQR with vertices P(4, 6), Q(-4, 2) and R(2, -6) and its image after a dilation with a scale factor of 0.5.

Dilations Solution d)

Similarity and Transformations I can perform and describe similarity transformations and prove figures are similar.

Similarity and Transformations Vocabulary (page 119 in Student Journal) similarity transformation: a dilation or a composition of rigid motions and dilations similar figures: have the same shape, but different sizes

Similarity and Transformations Example (space on page 119 in Student Journal) a) Graph segment AB with endpoints A(12, -6) and B(0, -3) and its image after the similarity transformation below. reflection in the y-axis and (x, y)  (1/3x, 1/3y)

Similarity and Transformations Solution a)

Similarity and Transformations b) Describe a similarity transformation that maps WXYZ to PQRS.

Similarity and Transformations Solution reflection in the x-axis and (x, y)  (2x, 2y)