Translations

New York State Common Core Math Geometry, Module 1, Lesson 16

Worksheets for Geometry

Student Outcomes

• Students learn the precise definition of a translation and perform a translation by construction.

Translations

Classwork

Exploratory Challenge

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.

1. Draw circle 𝑃: Center 𝑃, radius 𝐴𝐵.
2. Draw circle 𝐵: Center 𝐵, radius 𝐴𝑃.
3. Label the intersection of circle 𝑃 and circle 𝐵 as 𝑄.
4. Draw 𝑃𝑄.

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.

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Discussion

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:

1. For any point 𝑃 on the line 𝐴𝐵, 𝑇_𝐴𝐵 (𝑃) is the point 𝑄 on 𝐴𝐵 so that 𝑃𝑄 has the same length and the same direction as 𝐴𝐵 , and
2. For any point 𝑃 not on 𝐴𝐵 , 𝑇_𝐴𝐵 (𝑃) is the point 𝑄 obtained as follows. Let 𝑙 be the line passing through 𝑃 and parallel to ⃡𝐴𝐵 . Let 𝑚 be the line passing through 𝐵 and parallel to line 𝐴𝑃. The point 𝑄 is the intersection of 𝑙 and 𝑚.

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′.

Example 1

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.

Example 2

Use your compass and straightedge to apply 𝑇_𝐴𝐵 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.

Example 3

Use your compass and straightedge to apply 𝑇_𝐴𝐵 to △ 𝑃₁𝑃₂𝑃₃.

Relevant Vocabulary

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.

Lesson Summary

• A translation carries segments onto segments of equal length.
• A translation carries angles onto angles of equal measure.

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