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Learn about translations and vectors, properties of translations, how to use coordinate notation, and understand vectors in mathematics.

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Do Now

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  1. Do Now

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

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

  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. P' P THEOREM Q' Q USING PROPERTIES OF TRANSLATIONS THEOREM 9.1 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. P (2,4) The translation (x, y) (x + 4, y – 2) shifts each point 4 units to the right and 2 units down. P'(6,2) Q (1,2) Q'(5,0) (x, y) (x + a, y + b) Translations Using Coordinate Notation Translations in a coordinate plane can be described by the following coordinate notation: where a and b are constants. Each point shifts a units horizontally and b units vertically.

  8. 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). C' B' C(–1, 0) B(1, –1) A' C A(–1, –3) B ABC A'B'C' A Translations in a Coordinate Plane C'(–4, 4) B'(–2, 3) SOLUTION A'(– 4, –1) Plot original points. Shift each point 3 units to the left and 4 units up to translate vertices. A(–1, –3) A'(– 4, –1) B'(–2, 3) B(1, –1) C'(–4, 4) C(–1, 0)

  9. 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

  10. Q P The component form of a vector combines the horizontal and vertical components. So, the component form of PQ is 5, 3 . The vector is named PQ, which is read as “vector PQ.” The horizontal component of PQ is 5 and the vertical component is 3. 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. 3 units up 5 units to the right The diagram shows a vector. The initial point, or starting point, of the vector is P. The terminal point, or ending point, is Q.

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