Unit 2 Transformations / Rigid Motions Lesson 1: Reflections on the ...

[Pages:44]Unit 2

Transformations / Rigid Motions

Lesson 1: Reflections on the Coordinate Plane

Opening Exercise What do you remember about reflections??? Take the point (4, 2) and reflect it as stated. Plot the new point and state its coordinates.

Reflection in the x-axis

Reflection in the y-axis

Coordinates of the New Point: _____________ Coordinates of the New Point: _____________

Reflection in the line y = x

Reflection in the line x = 1

Coordinates of the New Point: _____________ Coordinates of the New Point: _____________

Summary of the Rules:

rx-axis : (x, y) ry-axis : (x, y) ry= x : (x, y)

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Vocabulary A transformation is a change in the position, shape, or size of a figure. A rigid motion is a transformation that changes only the position of the figure (length and angle measures are preserved). An image is the result of a transformation of a figure (called the pre-image). To identify the image of a point, use prime notation. The image of point A is A' (read as A prime). Example 1 Given ABC with vertices A (-5, 1), B (-1, 1) and C (-1, 7). a. Graph ABC on the axes provided below. b. On the same set of axes, graph A' B'C ' , the image of ABC reflected over the

x-axis. c. On the same set of axes, graph A'' B''C '' , the image of ABC reflected over the

y-axis.

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Example 2 Given ABC with vertices A (2, 3), B (0, 6) and C (2, 6). a. Graph ABC on the axes provided below. b. Graph and state the coordinates of A' B'C ' , the image of ABC reflected over the

line y = x. c. Graph and state the coordinates of A'' B''C '' , the image of A' B'C ' reflected over

the line y = -2 .

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Example 3 DOG has vertices D (-1, 1), O (-2, 5) and G (-5, 2) and D'O'G ' has vertices D' (3, -3), O' (7, -4) and G' (4, -7). a. Graph and label DOG and D'O'G ' b. Graph the line y = x - 2 c. What is the relationship between the line of reflection and the segments connecting

the corresponding points? (Think back to Unit 1)

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Homework

1. Using the point (-5, 3), find its image after the following reflections (the use of the grid is optional).

a.

rx - axis

b.

ry- axis

c.

ry= x

d.

rx = -1

e.

ry = 2

2. Given ABC with vertices A (2, 1), B (3, 4) and C (-4, 5). a. Graph ABC on the axes provided.

b. Graph and state the coordinates of A' B'C ' , the image of ABC reflected over the x-axis.

c. Graph and state the coordinates of A'' B''C '' , the image of A' B'C ' reflected over the line y = x .

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Lesson 2: Reflections off the Coordinate Plane

Opening Exercise You will need a compass and a straightedge As shown in the diagram to the right, ABC is reflected across DE and maps on to A' B'C ' . a. Use your straightedge to draw in

segments AA' , BB' and CC ' . b. Use your compass to measure the

distances from the pre-image point to DE and from the image point to DE. What do you notice about these distances?

c. What is the relationship between segment DE and each of the segments that were drawn in part a?

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Example 1 You will need a compass and a straightedge We now know that the line of reflection is the perpendicular bisector of the segments connecting the pre-image to the image point. We are going to use this, along with our knowledge of constructions, to construct the line of reflection. a. Connect any point to its image point. b. Draw the perpendicular bisector of this

segment.

This is the line of reflection! Each point and its image point are equidistant from this line!!! Selecting a second pair of points and constructing its perpendicular bisector can verify this.

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Exercises You will need a compass and a straightedge Construct the line of reflection for each image and its pre-image. 1.

2.

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