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

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.

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

USING PROPERTIES OF TRANSLATIONS Q ' P Q THEOREM THEOREM 9.1 Translation Theorem A translation is an isometry. Isometry--A transformation that preserves length and angle measure.

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

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.

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)

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

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.