WARM UP

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1. Graph ΔABC with the vertices A(–3, –2), B(4, 4), C(3, –3)

2. Graph ΔABC with the vertices D(1, 2), E(8, 8), F(7, 1)

Compare the two graphs and write a few sentences describing the similarities and differences of the two triangles.

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Unit 2-Lesson 1

Unit 2:Transformations

Lesson 1: Reflections and Translations

Objectives

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• I can identify and perform reflections and translations on a coordinate plane.

• I can predict the effect of a given ridged motion on a given figure.

• I can determine if two figures are congruent after a congruence transformation.

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Types of Transformations

Reflections: These are like mirror images as seen across a line or a point.

Translations ( or slides): This moves the figure to a new location with no change to the looks of the figure.

Rotations: This turns the figure clockwise or counter-clockwise but doesn’t change the figure.

Dilations: This reduces or enlarges the figure to a similar figure.

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Reflections

You could fold the picture along line l and the left figure would coincide with the corresponding parts of right figure.

l

You can reflect a figure using a line or a point. All measures (lines and angles) are preserved but in a mirror image.

Example: The figure is reflected across line l .

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Reflections – coordinates…

reflects across the y axis to line n

(2, 1) (-2, 1) & (5, 4) (-5, 4)

Reflection across the x-axis: the x values stay the same and the y values change sign. (x , y) (x, -y)

Reflection across the y-axis: the y values stay the same and the x values change sign. (x , y) (-x, y)

Example: In this figure, line l :

reflects across the x axis to line m.

(2, 1) (2, -1) & (5, 4) (5, -4)

ln

m

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Reflections across specific lines:

To reflect a figure across the line y = a or x = a, mark the corresponding points equidistant from the line.

i.e. If a point is 2 units above the line its corresponding image point must be 2 points below the line.

B(-3, 6) B′  (-3, -4)

C(-6, 2) C′  (-6, 0)

A(2, 3) A′ (2, -1).

Example:

Reflect the fig. across the line y = 1.

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Lines of Symmetry If a line can be drawn through a figure so the one side of the

figure is a reflection of the other side, the line is called a “line of symmetry.”

Some figures have 1 or more lines of symmetry. Some have no lines of symmetry.

One line of symmetry

Infinite lines of symmetry

Four lines of symmetry

Two lines of symmetry

No lines of symmetry

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• translation vector – shows direction and distance of the “slide”

VECTOR INTRODUCTION

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Copy the figure and given translation vector. Then draw the translation of the figure along the translation vector.

Step 1 Draw a line through each vertex parallel to vector .

Step 2 Measure the length ofvector . Locate point G'by marking off this distancealong the line throughvertex G, starting at G andin the same direction as thevector.

Step 3 Repeat Step 2 to locate points H', I', and J' to form the translated image.

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Translations (slides) If a figure is simply moved to another location without change to its

shape or direction, it is called a translation (or slide). A vector tells you how to translate a point <a, b> or <-a, -b>. If a point is moved “a” units to the right and “b” units up, then the

translated point will be at (x + a, y + b). If a point is moved “a” units to the left and “b” units down, then the

translated point will be at (x - a, y - b).

A

A′ 

Image A translates to image A′  by moving to the right 3 units and down 8 units.

Example:

A (2, 5) B (2+3, 5-8) A′ (5, -3)

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Translations in the Coordinate Plane

A. Graph ΔTUV with vertices T(–1, –4), U(6, 2), and V(5, –5) along the vector –3, 2.

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The vector indicates a translation 3 units left and 2 units up.

(x, y) → (x – 3, y + 2)

T(–1, –4) → (–4, –2)

U(6, 2) → (3, 4)

V(5, –5) → (2, –3)

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Translations in the Coordinate PlaneB. Graph pentagon PENTA with vertices P(1, 0), E(2, 2), T(4, –1), and A(2, –2) along the vector–5, –1.

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The vector indicates a translation 5 units left and 1 unit down.

(x, y) → (x – 5, y – 1)

P(1, 0) → (–4, –1)

E(2, 2) → (–3, 1)

N(4, 1) → (–1, 0)

T(4, –1) → (–1, –2)

A(2, –2) → (–3, –3)

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A. A'(–2, –5), B'(5, 1), C'(4, –6)

B. A'(–4, –2), B'(3, 4), C'(2, –3)

C. A'(3, 1), B'(–4, 7), C'(1, 0)

D. A'(–4, 1), B'(3, 7), C'(2, 0)

A. Graph ΔABC with the vertices A(–3, –2), B(4, 4), C(3, –3) along the vector –1, 3. Choose the correct coordinates for ΔA'B'C'.

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B. Graph □GHJK with the vertices G(–4, –2), H(–4, 3), J(1, 3), K(1, –2) along the vector 2, –2. Choose the correct coordinates for □G'H'J'K'.

A. G'(–6, –4), H'(–6, 1), J'(1, 1), K'(1, –4)

B. G'(–2, –4), H'(–2, 1), J'(3, 1), K'(3, –4)

C. G'(–2, 0), H'(–2, 5), J'(3, 5), K'(3, 0)

D. G'(–8, 4), H'(–8, –6), J'(2, –6), K'(2, 4)

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A. ANIMATION The graph shows repeated translations that result in the animation of the raindrop. Describe the translation of the raindrop from position 2 to position 3 in function notation and in words.