Chapter 4 4 1 Congruence and transformations SAT

Chapter 4 4 -1 Congruence and transformations

SAT Problem of the day

Objectives �Draw, identify, and describe transformations in the coordinate plane. �Use properties of rigid motions to determine whether figures are congruent and to prove figures congruent.

Transformations �What is a transformation? Answer: is a change in the position, shape, or size of a figure. �What are some types of transformations? �Answer: translations(slides), reflections, rotations and dilations

Example#1 �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) �translation 4 units left and 1 unit up

Example#2 �B. M: (x, y) → (x, -y) � A(1, 2), B(4, 2), C(3, 1) reflection across x-axis

Example#3 �. M: (x, y) → (y, -x) � R(-3, 0), E(-3, 3), C(-1, 3), T(-1, 0) 90°rotation clockwise with center of rotation (0, 0)

Example#4 �. M: (x, y) → (3 x, 3 y) � K(-2, -1), L(1, -1), N(1, -2)) dilation with scale factor 3 and center (0, 0)

Student guided practice �Do problems 3 -6 in your book page 220

Types of transformations �What is isometry ? �An isometry is a transformation that preserves length, angle measure, and area. Because of these properties, an isometry produces an image that is congruent to the preimage. �What is a rigid transformation? �A rigid transformation is another name for an isometry.

Transformations and congruence

Example#5 �Determine whether the polygons with the given vertices are congruent. �. A(-3, 1), B(2, 3), C(1, 1) � 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#6 � B. A(2, -2), B(4, -2), C(4, -4) � 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: (x, y) → (1. 5 x, 1. 5 y).

Student guided practice �Do problems 7 and 8 in your book page 220

Example#7 �Prove that the polygons with the given vertices are congruent. �A(1, 2), B(2, 1), C(4, 2) �P(-3, -2), Q(-2, -1), R(-3, 1) △ ABC can be mapped to △ A′B′C′ by a translation: (x, y) → (x – 3, y + 1); and then △ A′B′C′ can be mapped to △PQR by a rotation: (x, y) → (–y, x).

Example#8 �Prove that the polygons with the given vertices are congruent: A(-4, -2), B(-2, 1), C( 2, -2) and P(1, 0), Q(3, -3), R(3, 0). �The triangles are congruent because ABC can be mapped to A’B’C’ by a translation (x, y) → (x + 5, y + 2); and then A’B’C’ can be mapped to ABC by a reflection across the x-axis

Student guided practice �Do problemsd 9 and 10 in your book page 220

Architecture question? �Is there another transformation that can be used to create this frieze pattern? Explain your answer. Repeated reflections can create this frieze pattern; a reflection of any section over a line through either the left or right side of each section.

Homework!!! �Do problems even problems 13 -24 in your book page 220

Have a great day!!!