section 7.4 & 7.5 rigid motion in a plane translations and reflections glide reflections

Section 7.4 & 7.5

Rigid Motion in a PlaneTranslations and Reflections

Glide Reflections

Bell Work

J’( I,2), K’(4,1)L’(4,-3), M’(1,-3)

P’( -1,-3), Q’(-3,-5)R’(-4,-2), S’(-2,0)

D’( 4,1), E’(0,2)F’(2,5)

Outcomes

• You will be able to use vector notation to show translations.

• You will be able to identify what is a translation, a reflection, or a rotation.

• Identify and do a glide reflection or composition.

TranslationsA translation is a transformation which maps each point of a figure the same distance and in the same direction. The resulting figure after a transformation is called the image of the original figure, the preimage.

Theorem 7.4 - A translation is an Isometry.

Translations

Theorem 7.5

Theorem 7.5

ΔA’’B’’C’’

k and mAA’’, CC’’

2.8 in.

Yes, this is the definition of a reflection.

Example

• Sketch a triangle with vertices A (-1, -3), B (1, -1), C (-1, 0).• Then sketch the image of the triangle after the

translation (x, y) ⤍ (x-3, y+4)

Original Image A (-1,-3) A’( , )B (1, -1) B’( , )C (-1, 0) C’( , )

TranslationsEXAMPLE 1:ΔABC is translated 1 unit right and 4 units up. Draw the image ΔA’B’C’. What are the coordinates of: A (1, -3) A’ _________ B (3, 0) B’ _________ C (4, -2) C’_________

ΔABC ΔA’B’C’ This translation can be written as (x, y) (x , y ).

TranslationsEXAMPLE 2: ΔJKL has coordinates J (0,2), K (3,4), and L (5,1). a) Draw ΔJKL.b) Draw the image ΔJ’K’L’ after a translation of 4 units to the left and 5 units up. Label the triangle. What are the coordinates of:

J (0, 2) J’ _________ K (3, 4) K’ _________

L (5, 1) L’__________ Rule: (x, y) ( , ) Are the figures congruent or similar? Explain how you know.

TranslationExample 3: Write a general rule which describes the translation shown below. ΔLMN is the preimage.

(x, y) ( , )

TranslationsEXAMPLE 4:a) Graph points T(0,3), U(2, 4) and V(5, -1) and connect the points to make a triangle.b) Translate ΔTUV using the rule (x, y) (x - 3, y - 1).c) In words, describe what the rule says.

d) Draw the image ΔT’U’V’.e) Identify the coordinates of ΔT’U’V’.

T’U’V’

f) Using the image of ΔT’U’V’ perform an additional translation using the rule (x, y) (x + 3, y - 3). State the new coordinates of ΔT”U”V”. Is this new image congruent or similar to the preimage?

1 2 3 4 5–1–2–3–4–5 x

1

2

3

4

5

–1

–2

–3

–4

–5

y

Vectors

• Vector: is a quantity that has both direction and magnitude, and is represented by an arrow drawn between two points.

• Symbol PQ• Component form: ⟨3, 4⟩• Arrow Notation: (x, y) ⟶ (x+3, y+4)

A) AS = ⟨ 4,-4 ⟩⇀B) MN = ⟨ 0, 4 ⟩⇀

⇀⇀

Section 7.5 Glide Reflections and Compositions

When a translation and a reflection are done one after the other it is known as a glide reflection.

A glide reflection is a transformation in which every point P is mapped onto a point P’’ by the following steps:1. A translation maps P onto P’.

2. A reflection in a line k parallel to the direction of the translation maps P’ onto P’’.

**As long as the line of reflection is parallel to the direction of the translation, it does not matter whether you glide first and then reflect, or reflect first and then glide!**

Glide Reflections

Glide ReflectionUse the information to sketch the

image if ΔABC after a glide reflection,

A (-1, -3), B(-4, -1), and C (-6,-4)• Translation: (x,y) ⤍ (x + 10, y)• Reflection: in the x-axis A’’ ( , ) , B’’ ( , ), C’’ ( , )

If we reversed the order of the transformations (reflection then translation), will ΔA’’B’’C’’ have the same coordinates found in the example?

Yes, because the line of reflection is parallel to the direction of the translation!

Compositions

When two or more transformations are combined to produce a single transformation, the result is called a composition.

Theorem 7.6 Composition Theorem:The composition of two, or more, isometries is

an Isometry.

CompositionsSketch the image of PQ after a composition of the given rotation and reflection: P (2, -2), Q (3, -4)

Rotation: 90 degrees CCW about the origin

Reflection: in the y-axis

Repeat the exercise, but switch the order of the composition; reflection then rotation.What do you notice?

CompositionsSketch the image of PQ after a composition of the given reflection and reflection: P (2, -2), Q (3, -4)

Reflection: in the y-axis

Rotation: 90 degrees CCW about the origin

Compositions

• Look over the IP and ask questions