Question

given two parallel lines, explain how to transform one onto the other using the following.
a. a rotation. describe the center and an angle of rotation.
b. a translation. describe all possible translations.
a. choose the correct answer below.
o a. determine the center of rotation by finding the midpoint of a segment whose endpoints are on the two given lines. then perform a quarter - turn rotation about this point.
o b. determine the center of rotation by finding the midpoint of a segment whose endpoints are on the two given lines. then perform a half - turn rotation about this point.
o c. determine the center of rotation by finding the midpoint of a segment whose endpoints are between the two given lines. then perform a half - turn rotation about this point.
o d. determine the center of rotation by finding the midpoint of a segment whose endpoints are between the two given lines. then perform a quarter - turn rotation about this point.
b. choose the correct answer below.
o a. given lines k and m, and if k⊥m, a and b are two different points on k, then the image of k under the translation by \\( \overrightarrow{ab} \\) is m.
o b. given lines k and m, and if k∥m, a is any point on k, and b is on m, then the image of k under the translation by \\( \overrightarrow{ab} \\) is m.
o c. given lines k and m, and if k∥m and a and b are two different points on k, then the image of k under the translation by \\( \overrightarrow{ab} \\) is m.
o d. given lines k and m, and if k⊥m, a is any point on k, and b is on m, then the image of k under the translation by \\( \overrightarrow{ab} \\) is m.

Explanation:

Step1: Analyze translation for parallel lines

For two parallel lines \(k\) and \(m\), a translation vector \(\overrightarrow{AB}\) (where \(A\) is on \(k\) and \(B\) is on \(m\)) will map \(k\) to \(m\). If \(k\parallel m\), the translation by \(\overrightarrow{AB}\) (where \(A\) and \(B\) are points on \(k\) and \(m\) respectively) is valid. If \(k\perp m\), a translation by \(\overrightarrow{AB}\) (where \(A\) is on \(k\) and \(B\) is on \(m\)) does not map \(k\) to \(m\) because the direction of translation is not consistent with the relationship between perpendicular lines.

Step2: Analyze rotation for parallel lines

For a rotation, the center of rotation is the mid - point of a segment whose endpoints are on the two parallel lines. A half - turn (rotation by \(180^{\circ}\)) about the mid - point of a segment with endpoints on \(k\) and \(m\) will map \(k\) to \(m\) (since for parallel lines, a \(180^{\circ}\) rotation about the mid - point of a segment connecting them is a valid transformation). A quarter - turn (rotation by \(90^{\circ}\)) or other non - \(180^{\circ}\) rotations (in the context of mapping one parallel line to another) about a point not following the mid - point rule (for the segment between the lines) will not map \(k\) to \(m\)

Answer:

a. For a rotation to transform one parallel line to another:

• The center of rotation is the mid - point of a segment whose endpoints are on the two given lines. The angle of rotation is \(180^{\circ}\) (a half - turn).

b. For a translation to transform one parallel line to another:

• If \(k\parallel m\), and \(A\) is a point on \(k\) and \(B\) is a point on \(m\), then the translation by \(\overrightarrow{AB}\) maps \(k\) to \(m\). All possible translations are of the form \(\overrightarrow{PQ}\), where \(P\) is on \(k\) and \(Q\) is on \(m\) and \(\overrightarrow{PQ}\) is parallel to the direction of the parallel lines (the set of all such vectors \(\overrightarrow{PQ}\) forms a translation that maps \(k\) to \(m\))