In Lesson 4, you completed a construction exercise that resulted in a pair of parallel lines (Problem 1 from the Problem Set). Now we examine an alternate construction.
Construct the line parallel to a given line 𝐴𝐵 through a given point 𝑃.
Note: Circles 𝑃 and 𝐵 intersect in two locations. Pick the intersection 𝑄 so that points 𝐴 and 𝑄 are in opposite halfplanes of line 𝑃𝐵.
To perform a translation, we need to use the previous construction. Let us investigate the definition of translation.
For vector 𝐴𝐵 , the translation along 𝐴𝐵 is the transformation 𝑇𝐴𝐵 of the plane defined as follows:
Note: The parallel line construction on the previous page shows a quick way to find the point 𝑄 in part 2 of the definition of translation.
In the figure to the right, quadrilateral 𝐴𝐵𝐶𝐷 has been translated the length and direction of vector 𝐶𝐶′. Notice that the distance and direction from each vertex to its corresponding vertex on the image are identical to that of 𝐶𝐶′.
Draw the vector that defines each translation below.
Finding the vector is relatively straightforward. Applying a vector to translate a figure is more challenging. To translate a figure, we must construct parallel lines to the vector through the vertices of the original figure and then find the points on those parallel lines that are the same direction and distance away as given by the vector.
Use your compass and straightedge to apply 𝑇𝐴𝐵 to segment 𝑃1𝑃2 . Note: Use the steps from the Exploratory Challenge twice for this question, creating two lines parallel to 𝐴𝐵: one through 𝑃1 and one through 𝑃2.
Use your compass and straightedge to apply 𝑇𝐴𝐵 to △ 𝑃1𝑃2𝑃3.
PARALLEL: Two lines are parallel if they lie in the same plane and do not intersect. Two segments or rays are parallel if the lines containing them are parallel lines.
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