Unit 1, Lesson 12
Defining Translations
Integrated Math 1
12.1
Warm-up
What do you notice? What do you wonder?
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What do you notice? What do you wonder?
A translation slides a figure a given distance in a given direction with no rotation. The distance and direction are given by a directed line segment. The arrow of the directed line segment specifies the direction of the translation, and the length of the directed line segment specifies how far the figure gets translated.
More precisely, a translation of a point by a directed line segment is a transformation that takes to so that the directed line segment is parallel to , goes in the same direction as , and is the same length as .
Directed line segment T, slants upward and to the right, arrow at top end. Directed line segment A A prime, parallel and congruent to T, slants upward and to the right. Endpoint on bottom end, A, arrow at top end touching A prime.
Here is a translation of 3 points. Notice that the directed line segments , , and are each parallel to , go in the same direction as , and are the same length as .
Triangle C D E and a translation of three points. Directed line segments C C prime, D D prime, E E prime and v are all the same length and direction, slanting slightly downward and to the right, with the endpoint on the top end and the arrow at the bottom end. Triangle C prime D prime E prime at the end of three directed line segments.
Also notice that segment is parallel to segment . We proved that this would always be true, so we can write a theorem that says translations take lines to parallel lines or to themselves. A theorem is a statement that has been proved mathematically.
A directed line segment is a line segment that has distance (length) and direction.
The arrow on this directed line segment shows that it starts at point and ends at point .
A theorem is a statement that has been proved mathematically.
A translation is a rigid transformation that is defined by a directed line segment. It takes one point to another point so that:
In the figure shown here, is the image of under the translation given by the directed line segment .