Definition: Let \vec a be a vector and М a geometric figure in the plane. A new geometric figure М_1 is translation or parallel shift of М by the vector \vec a if every point P of М corresponds to a point P_1 of М_1 such that the vector \overrightarrow {MM_1} = \vec a . Example: In the interactivity below triangle DEF is a translation of triangle ABC by the vector \vec a . Interactivity 1: Translation Directions for interactivity Click and drag the blue vertex points to change the triangle. Click and drag the blue end point of the vector \vec a to change it. Select the checkboxes as desired to see the translations. If you have GeoGebra installed, double-click on interactivity to open it on your computer. |

Regulation: If М_1 is a translation of М by the vector \vec a , then М and М_1 are congruent, that is М \cong М_1 . From this it follows that М and М_1 have the same geometric properties such as perimeter, area, etc. |

Definition 2: Let Т=(x,y) be an ordered pair of numbers. A polygon N is a translation of another polygon M by Т if M and N have the same number of vertices and if the coordinates of every vertex of N can be obtained by adding T to a corresponding vertex of M. Example: Let Т=(12,-2) and let M be the quadrilateral with vertices: (-4,4), (3,4), (1,11) и (-4,7). Then the quadrilateral N with vertices: (-4+12,4-2)=(8,2), (3+12,4-2)=(15,2), (1+12,11-2)=(13,9) и (-4+12,7-2)=(8,5) is a translation of M by T. Interactivity 2: Translation Directions for interactivity Click and drag any of the blue points to form a quadrilateral of your choice. Click and drag the sliders Tx and Ty to change the translation vector T. Try to determine the coordinates of the vertices of the translation. Select the checkbox to see the translation and check your results. To zoom, right-click on any empty space in the interactivity and select "zoom" from the menu.
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Regulation: Let Т=(x,y) . Define \vec a to be the radius-vector starting at (0,0) and ending at T, that is: \vec a =\{ x,y \} . Then М_1 is the translation of М by the vector \vec a if М_1=М+\{ x,y \} . |

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