Transcript Slide 1
1-7 Transformations in the Coordinate Plane Objectives Identify reflections, rotations, and translations. Graph transformations in the coordinate plane. Holt McDougal Geometry 1-7 Transformations in the Coordinate Plane Vocabulary transformation preimage image Holt McDougal Geometry reflection rotation translation 1-7 Transformations in the Coordinate Plane A transformation is a change in the position, size, or shape of a figure. The original figure is called the preimage. The resulting figure is called the image. Arrow notation () is used to describe a transformation, and primes (’) are used to label the image. Holt McDougal Geometry 1-7 Transformations in the Coordinate Plane Holt McDougal Geometry 1-7 Transformations in the Coordinate Plane Example 1A: Identifying Transformation Identify the transformation. Then use arrow notation to describe the transformation. The transformation cannot be a reflection because each point and its image are not the same distance from a line of reflection. 90° rotation, ∆ABC ∆A’B’C’ Holt McDougal Geometry 1-7 Transformations in the Coordinate Plane Example 1B: Identifying Transformation Identify the transformation. Then use arrow notation to describe the transformation. The transformation cannot be a translation because each point and its image are not in the same relative position. reflection, DEFG D’E’F’G’ Holt McDougal Geometry 1-7 Transformations in the Coordinate Plane Check It Out! Example 1 Identify each transformation. Then use arrow notation to describe the transformation. a. translation; MNOP M’N’O’P’ Holt McDougal Geometry b. Counterclockwise rotation; ∆XYZ ∆X’Y’Z’ 1-7 Transformations in the Coordinate Plane Example 2: Drawing and Identifying Transformations A figure has vertices at A(1, –1), B(2, 3), and C(4, –2). After a transformation, the image of the figure has vertices at A'(–1, –1), B'(–2, 3), and C'(–4, –2). Draw the preimage and image. Then identify the transformation. Plot the points. Then use a straightedge to connect the vertices. The transformation is a reflection across the y-axis because each point and its image are the same distance from the y-axis. Holt McDougal Geometry 1-7 Transformations in the Coordinate Plane Check It Out! Example 2 A figure has vertices at E(2, 0), F(2, -1), G(5, -1), and H(5, 0). After a transformation, the image of the figure has vertices at E’(0, 2), F’(1, 2), G’(1, 5), and H’(0, 5). Draw the preimage and image. Then identify the transformation. Plot the points. Then use a straightedge to connect the vertices. The transformation is a 90° counterclockwise rotation with rotation center at origin O(0,0). Holt McDougal Geometry What happens when we translate a shape ? Transformations in the Coordinate Plane 1-7 Transformations The shape remains the same size and shape and the same way up – it just……. slides . 3. Translation Horizontal translation Vertical translation Write the rule to describe a translation from…….. D 1. A to B 2. A to D 3. B to C 4. D to C Holt McDougal Geometry C A B 1-7 Transformations in the Coordinate Plane Writing a Rule for a translation y B’ B A’ A C’ C x Holt McDougal Geometry To write a rule, look for the change in the x and y values for a coordinate. From A to A’. The point has gone 3 units to the right and 2 units up. The rule is (x,y) → (x ± ?), (y ± ?) The rule is (x,y) → (x +3), (y +2) 1-7 Transformations in the Coordinate Plane Your turn, Write the Rule. y The rule is (x,y) → (x ± ?), (y ± ?) B The rule is (x,y) → (x -4), (y -4) B’ A A ’ C’ Holt McDougal Geometry C x 1-7 Transformations in the Coordinate Plane Your turn, Write the Rule. y The rule is (x,y) → (x ± ?), (y ± ?) B A A ’ Holt McDougal Geometry B’ The rule is (x,y) → (x +1), (y - 5) C C’ x 1-7 Transformations in the Coordinate Plane Your turn, Write the Rule. y B B’ The rule is (x,y) → (x ± ?), (y ± ?) A A ’ The rule is (x,y) → (x - 5), (y - 2) C C’ x Holt McDougal Geometry 1-7 Transformations in the Coordinate Plane Example 3: Translations in the Coordinate Plane Find the coordinates for the image of ∆ABC after the translation (x, y) (x + 2, y - 1). Draw the image. Step 1 Find the coordinates of ∆ABC. The vertices of ∆ABC are A(–4, 2), B(–3, 4), C(–1, 1). Holt McDougal Geometry 1-7 Transformations in the Coordinate Plane Example 3 Continued Step 2 Apply the rule (x, y) (x + 2, y - 1) to find the vertices of the image. A’(–4 + 2, 2 – 1) = A’(–2, 1) B’(–3 + 2, 4 – 1) = B’(–1, 3) C’(–1 + 2, 1 – 1) = C’(1, 0) Step 3 Plot the points. Then finish drawing the image by using a straightedge to connect the vertices. Holt McDougal Geometry 1-7 Transformations in the Coordinate Plane To find coordinates for the image of a figure in a translation, add a to the x-coordinates of the preimage and add b to the y-coordinates of the preimage. Translations can also be described by a rule such as (x, y) (x + a, y + b). Holt McDougal Geometry 1-7 Transformations in the Coordinate Plane Check It Out! Example 3 Find the coordinates for the image of JKLM after the translation (x, y) (x – 2, y + 4). Draw the image. Step 1 Find the coordinates of JKLM. The vertices of JKLM are J(1, 1), K(3, 1), L(3, –4), M(1, –4), . Holt McDougal Geometry 1-7 Transformations in the Coordinate Plane Check It Out! Example 3 Continued Step 2 Apply the rule to find the vertices of the image. J’(1 – 2, 1 + 4) = J’(–1, 5) J’ K’ K’(3 – 2, 1 + 4) = K’(1, 5) L’(3 – 2, –4 + 4) = L’(1, 0) M’(1 – 2, –4 + 4) = M’(–1, 0) Step 3 Plot the points. Then finish drawing the image by using a straightedge to connect the vertices. Holt McDougal Geometry M’ L’ 1-7 Transformations in the Coordinate Plane Example 4: Art History Application The figure shows part of a tile floor. Write a rule for the translation of hexagon 1 to hexagon 2. Step 1 Choose two points. Choose a Point A on the preimage and a corresponding Point A’ on the image. A has coordinate (2, –1) and A’ has coordinates Holt McDougal Geometry A’ A 1-7 Transformations in the Coordinate Plane Example 4 Continued The figure shows part of a tile floor. Write a rule for the translation of hexagon 1 to hexagon 2. Step 2 Translate. To translate A to A’, 3 units are subtracted from the xcoordinate and 1 units are added to the y-coordinate. Therefore, the translation rule is (x, y) → (x – 3, y + 1 ). Holt McDougal Geometry A’ A 1-7 Transformations in the Coordinate Plane Check It Out! Example 4 Use the diagram to write a rule for the translation of square 1 to square 3. Step 1 Choose two points. Choose a Point A on the preimage and a corresponding Point A’ on the image. A has coordinate (3, 1) and A’ has coordinates (–1, –3). Holt McDougal Geometry A’ 1-7 Transformations in the Coordinate Plane Check It Out! Example 4 Continued Use the diagram to write a rule for the translation of square 1 to square 3. Step 2 Translate. To translate A to A’, 4 units are subtracted from the x-coordinate and 4 units are subtracted from the y-coordinate. Therefore, the translation rule is (x, y) (x – 4, y – 4). Holt McDougal Geometry A’ 1-7 Transformations in the Coordinate Plane Lesson Quiz: Part I 1. A figure has vertices at X(–1, 1), Y(1, 4), and Z(2, 2). After a transformation, the image of the figure has vertices at X'(–3, 2), Y'(–1, 5), and Z'(0, 3). Draw the preimage and the image. Identify the transformation. translation 2. What transformation is suggested by the wings of an airplane? reflection Holt McDougal Geometry 1-7 Transformations in the Coordinate Plane Lesson Quiz: Part II 3. Given points P(-2, -1) and Q(-1, 3), draw PQ and its reflection across the y-axis. 4. Find the coordinates of the image of F(2, 7) after the translation (x, y) (x + 5, y – 6). (7, 1) Holt McDougal Geometry 1-7 Transformations in the Coordinate Plane Architecture Application 5. Is there another transformation that can be used to create this frieze pattern? Explain your answer. Holt McDougal Geometry 1-7 Transformations in the Coordinate Plane Objectives Use properties of rigid motions to determine whether figures are congruent. Holt McDougal Geometry 1-7 Transformations in the Coordinate Plane Vocabulary Isometry Rigid transformation Dilation Holt McDougal Geometry 1-7 Transformations in the Coordinate Plane 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. A rigid transformation is another name for an isometry. Reflection, rotation and translation are isometry, or rigid transformation. Holt McDougal Geometry 1-7 Transformations in the Coordinate Plane A dilation with scale factor k > 0 and center (0, 0) maps (x, y) to (kx, ky). Dilation is not isometry. It is not a rigid transformation. Holt McDougal Geometry 1-7 Transformations in the Coordinate Plane Example 5: Drawing and Identifying Transformations M: (x, y) → (3x, 3y) K(-2, -1), L(1, -1), N(1, -2)) dilation with scale factor 3 and center (0, 0) Holt McDougal Geometry 1-7 Transformations in the Coordinate Plane Check It Out! Example 5 1. Apply the transformation M : (x, y) →(3x, 3y) to the polygon with vertices D(1, 3), E(1, -2), and F(3, 0). Name the coordinates of the image points. Identify and describe the transformation. D’(3, 9), E’(3, -6), F’(9, 0); dilation with scale factor 3 Holt McDougal Geometry 1-7 Transformations in the Coordinate Plane Holt McDougal Geometry 1-7 Transformations in the Coordinate Plane Holt McDougal Geometry 1-7 Transformations in the Coordinate Plane Lesson Quiz : Part-I Apply the transformation M to the polygon with the given vertices. Identify and describe the transformation. 1. M: (x, y) → (3x, 3y) A(0, 1), B(2, 1), C(2, -1) dilation with scale factor 3 and center (0, 0) 2. M: (x, y) → (-y, x) A(0, 3), B(1, 2), C(4, 5) 90° rotation counterclockwise with center of rotation (0, 0) Holt McDougal Geometry 1-7 Transformations in the Coordinate Plane Lesson Quiz : Part-II 3. M: (x, y) → (x + 1, y - 2) A(-2, 1), B(-2, 4), C(0, 3) translation 1 unit right and 2 units down 4. Determine whether the triangles are congruent. A(1, 1), B(1, -2), C(3, 0) J(2, 2), K(2, -4), L(6, 0) not ≌; △ ABC can be mapped to △ JKL by a dilation with scale factor k ≠ 1: (x, y) → (2x, 2y). Holt McDougal Geometry 1-7 Transformations in the Coordinate Plane Lesson Quiz : Part-III 5. Prove that the triangles are congruent. A(1, -2), B(4, -2), C(1, -4) D(-2, 2), E(-5, 2), F(-2, 0) △ ABC can be mapped to △ A′B′C′ by a translation: (x, y) → (x + 1, y + 4); and then △ A′B′C′ can be mapped to △DEF by a reflection: (x, y) → (-x, y). Holt McDougal Geometry