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More Lessons for Geometry

Common Core 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 π΄π΅ through a given point π.

- Draw circle π: Center π, radius π΄π΅.
- Draw circle π΅: Center π΅, radius π΄π.
- Label the intersection of circle π and circle π΅ as π.
- Draw ππ.

Note: Circles π and π΅ intersect in two locations. Pick the intersection π so that points π΄ and π are in opposite halfplanes of line ππ΅.

**Discussion**

To perform a translation, we need to use the previous construction. Let us investigate the definition of translation.

For vector π΄π΅ , the translation along π΄π΅ is the transformation ππ΄π΅ of the plane defined as follows:

- For any point π on the line π΄π΅, π
_{π΄π΅}(π) is the point π on π΄π΅ so that ππ has the same length and the same direction as π΄π΅ , and - For any point π not on π΄π΅ , π
_{π΄π΅}(π) is the point π obtained as follows. Let π be the line passing through π and parallel to β‘π΄π΅ . Let π be the line passing through π΅ and parallel to line π΄π. The point π is the intersection of π and π.

Note: The parallel line construction on the previous page shows a quick way to find the point π in part 2 of the definition of translation.

In the figure to the right, quadrilateral π΄π΅πΆπ· has been translated the length and direction of vector πΆπΆβ². Notice that the distance and direction from each vertex to its corresponding vertex on the image are identical to that of πΆπΆβ².

**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 π_{π΄π΅}
to segment π_{1}π_{2}
.
Note: Use the steps from the Exploratory Challenge twice for this question,
creating two lines parallel to π΄π΅: one through π_{1} and one through π_{2}.

**Example 3**

Use your compass and straightedge to apply π_{π΄π΅} to β³ π_{1}π_{2}π_{3}.

**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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