Translations

1
Lesson 7.4

• Translations
• and
• Vectors

2
Lesson 7.4

• Define a translation
• Identify a translation in a plane
• Use vectors to describe a translation
• Identify vector notation

3
Translation Definition

• A translation is a transformation that maps an
  object by shifting or sliding the object and all
  of its parts in a straight light.
• A translation must also move the entire object
  the same distance.

4
Theorem 7.4Translation Theorem

• A translation is an isometry.

5
Theorem 7.5Distance of Translation Theorem

• If lines k and m are parallel, then a reflection
  in line k followed by a reflection in line m is a
  translation.
• The segment between P and P is perpendicular to
  k and m.
• The distance of the translation is twice the
  distance between the reflecting lines.

x
2x
6
Coordinate form

• Every translation has a horizontal movement and a
  vertical movement.
• A translation can be described in coordinate
  notation.
• (x,y) ? (xa , yb)
• Which tells you to move a units horizontal and b
  units vertical.

Q
b units up
P
a units to the right
7
Vectors

• Another way to describe a translation is to use a
  vector.
• A vector is a quantity that shows both direction
  and magnitude, or size.
• It is represented by an arrow pointing from
  pre-image to image.
• The starting point at the pre-image is called the
  initial point.
• The ending point at the image is called the
  terminal point.

8
Component Form of Vectors

• Component form of a vector is a way of combining
  the individual movements of a vector into a more
  simple form.
• ?x , y?
• Naming a vector is the same as naming a ray.
• PQ

Q
y units up
P
x units to the right
9
Use of Vectors

• Adding/subtracting vectors
• Add/subtract x values and then add y values
• ?2 , 6? ?3 , -4?
• ?5 , 2?
• Distributive property of vectors
• Multiply each component by the constant
• 5 ?3 , -4?
• ?15 , -20?
• Length of vector
• Pythagorean Theorem
• x2 y2 length2
• Direction of vector
• Inverse tangent
• tan-1 (y/x)