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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. |
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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. |
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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 on the text Т and the drag your mouse up/down and left/right to change 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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