1. Do Now

2. Section 9.1 – Translate Figures and Use Vectors
Chapter 9
Section 9.1 – Translate Figures and Use Vectors

3. USING PROPERTIES OF TRANSLATIONS
Q ' P Q
A translation is a transformation that maps every two points P and Q in the plane to points P ' and Q ' , so that the following properties are true:
PP ' = QQ '
PP ' QQ ' , or PP ' and QQ ' are collinear.

4. Intro: 3 Types of Transformations
Reflection Rotation Translation
in a line about a point F F F F F F
Preimage: original figure
Image: new figure after transformation

5. USING PROPERTIES OF TRANSLATIONS
Q ' P Q
THEOREM
THEOREM Translation Theorem
A translation is an isometry.
Isometry--A transformation that preserves length and angle measure.

6. USING PROPERTIES OF TRANSLATIONS
You can find the image of a translation by gliding a figure in the plane.

7. Translations Using Coordinate Notation
P (2,4)
Q (1,2)
The translation (x, y) (x + 4, y – 2) shifts each point 4 units to the right and 2 units down.
P ' (6,2)
Q ' (5,0)
Translations in a coordinate plane can be described by the following coordinate notation:
(x, y) (x + a, y + b)
where a and b are constants. Each point shifts a units horizontally and b units vertically.

8. Translations in a Coordinate Plane
Sketch a triangle with vertices A(–1, –3), B(1, –1), and C(–1, 0). Then sketch the image of the triangle after the translation (x,y) (x – 3, y + 4).
A(–1, –3)
B(1, –1)
C(–1, 0)
C ' (– 4, 4)
B' (–2, 3)
A '
C '
B '
SOLUTION
A' (– 4, –1)
Plot original points. Shift each point 3 units to the left and 4 units up to translate vertices. A C B
ABC
 A'B 'C '
A(–1, –3)
A' (– 4, –1)
B(1, –1)
B' (–2, 3)
C(–1, 0)
C ' (– 4, 4)

10. Vectors
Vector: quantity that shows direction and magnitude (size) and is represented by an arrow drawn between two points.
Initial Point: starting point of vector
Terminal Point: ending point of vector
Component form of a vector: combines the horizontal and vertical components and is represented:
Terminal point
h: horizontal component
v: vertical component v
Initial point h

11. TRANSLATIONS USING VECTORS
Another way to describe a translation is by using a vector. A vector is a quantity that has both direction and magnitude, or size, and is represented by an arrow drawn between two points. P Q
3 units up
5 units
to the right
The diagram shows a vector.
The component form of a vector combines the horizontal and vertical components. So, the component form of PQ is 5, 3 .
The initial point, or starting point, of the vector is P.
The terminal point, or ending point, is Q.
The vector is named PQ, which is read as “ vector PQ.”
The horizontal component of PQ is 5 and the vertical component is 3.