identify reflections, rotations, and translations. graph transformations in the coordinate plane

Holt McDougal Geometry

1-7 Transformations in the Coordinate Plane

Identify reflections, rotations, and translations. Graph transformations in the coordinate plane.

Objectives



transformation reflectionpreimage rotationimage translation

Vocabulary



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.


1-7 Transformations in the Coordinate PlaneExample 1A: Identifying Transformation

Identify the transformation. Then use arrow notation to describe the transformation.

The transformation cannotbe a reflection because eachpoint and its image are not the same distance from a line of reflection.

90° rotation, ∆ABC ∆A’B’C’


1-7 Transformations in the Coordinate PlaneExample 1B: Identifying Transformation

Identify the transformation. Then use arrow notation to describe the transformation.

The transformation cannot bea translation because eachpoint and its image are not inthe same relative position.

reflection, DEFG D’E’F’G’


1-7 Transformations in the Coordinate PlaneCheck It Out! Example 1

Identify each transformation. Then use arrow notation to describe the transformation.

translation; MNOP M’N’O’P’ Counterclockwise rotation; ∆XYZ ∆X’Y’Z’

a. b.


1-7 Transformations in the Coordinate PlaneExample 2: Drawing and Identifying TransformationsA 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.



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).


1-7 Transformations in the Coordinate PlaneTransformations

3. Translation

Horizontal translation

Vertical translation

What happens when we translate a shape ?The shape remains the same size and shape

and the same way up – it just……. .slides

Write the rule to describe a translationfrom……..

1. A to B

2. A to D

3. B to C

4. D to C

CD

A B



Writing a Rule for a translation

A

B

C

x

y

A’

B’

C’

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 PlaneYour turn, Write the Rule.

x

y

A

B

C

The rule is (x,y) → (x ± ?), (y ± ?)

The rule is (x,y) → (x -4), (y -4)

C’

B’

A’



x

y

A

B

C

The rule is (x,y) → (x ± ?), (y ± ?)

The rule is (x,y) → (x +1), (y - 5)

C’

B’

A’



x

y

A

B

C

The rule is (x,y) → (x ± ?), (y ± ?)

The rule is (x,y) → (x - 5), (y - 2)

C’

B’

A’



Find the coordinates for the image of ∆ABC after the translation (x, y) (x + 2, y - 1). Draw the image.

Example 3: Translations in the Coordinate Plane

Step 1 Find the coordinates of ∆ABC.

The vertices of ∆ABC are A(–4, 2), B(–3, 4), C(–1, 1).


1-7 Transformations in the Coordinate PlaneExample 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.



To find coordinates for the image of a figure in a translation, add a to thex-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).



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), .


1-7 Transformations in the Coordinate PlaneCheck 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)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.

J’ K’

M’ L’


1-7 Transformations in the Coordinate PlaneExample 4: Art History Application

The figure shows part of a tile floor. Write a rule for the translation of hexagon 1 to hexagon 2.

A

A’

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


1-7 Transformations in the Coordinate PlaneExample 4 Continued

The figure shows part of a tile floor. Write a rule for the translation of hexagon 1 to hexagon 2.

A

A’

Step 2 Translate. To translate A to A’, 3 units are subtracted from the x-coordinate and 1 units are added to the y-coordinate. Therefore, the translation rule is (x, y) → (x – 3, y + 1 ).



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). A’


1-7 Transformations in the Coordinate PlaneCheck 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).

A’


1-7 Transformations in the Coordinate PlaneLesson 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


1-7 Transformations in the Coordinate PlaneLesson 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)


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.



Use properties of rigid motions to determine whether figures are congruent.

Objectives



IsometryRigid transformationDilation

Vocabulary



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.



A dilation with scale factor k > 0 andcenter (0, 0) maps (x, y) to (kx, ky).

Dilation is not isometry. It is not a rigid transformation.


1-7 Transformations in the Coordinate PlaneExample 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)



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


1-7 Transformations in the Coordinate PlaneLesson Quiz : Part-I

Apply the transformation M to the polygon with the given vertices. Identify and describe the transformation.

dilation with scale factor 3 and center (0, 0)

1. M: (x, y) → (3x, 3y)A(0, 1), B(2, 1), C(2, -1)

2. M: (x, y) → (-y, x)A(0, 3), B(1, 2), C(4, 5)

90° rotation counterclockwise with center of rotation (0, 0)


1-7 Transformations in the Coordinate PlaneLesson Quiz : Part-II

translation 1 unit right and 2 units down

3. M: (x, y) → (x + 1, y - 2)A(-2, 1), B(-2, 4), C(0, 3)

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).


1-7 Transformations in the Coordinate PlaneLesson Quiz : Part-III

△ 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).

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)

identify reflections, rotations, and translations. graph transformations in the coordinate plane

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