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.
b. 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.
a. choose the correct answer below.
o a. 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 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:

Part a

When dealing with two parallel lines, a translation can map one line onto the other. The vector of translation can be determined by the segment connecting a point on one line to its corresponding point on the other line. For example, if we have two parallel lines \(k\) and \(m\), and we take a point \(A\) on \(k\) and a point \(B\) on \(m\), the translation by the vector \(\overrightarrow{AB}\) will map line \(k\) onto line \(m\) (assuming \(k\parallel m\)).

Part b

When two lines are perpendicular (\(k\perp m\)), a rotation of \(180^{\circ}\) (a half - turn) about the mid - point of a segment whose endpoints are on the two lines can map one line onto the other. Let \(A\) be a point on \(k\) and \(B\) be a point on \(m\). The mid - point \(O\) of segment \(AB\) is the center of rotation. A rotation of \(180^{\circ}\) about point \(O\) will map line \(k\) onto line \(m\) (since for a \(180^{\circ}\) rotation, if \(O\) is the center, and \(A\) is a point on \(k\), its image \(A'\) after a \(180^{\circ}\) rotation about \(O\) satisfies \(OA = OA'\) and \(\angle AOA'=180^{\circ}\). If \(k\perp m\), this rotation will map the direction of \(k\) to the direction of \(m\)).

Answer:

a. A translation. To perform the translation, find the vector between a point on one line and its corresponding point on the other line. For example, if \(A\) is on \(k\) and \(B\) is on \(m\) (\(k\parallel m\)), the translation by \(\overrightarrow{AB}\) maps \(k\) to \(m\).
b. A half - turn ( \(180^{\circ}\) rotation). To perform the rotation, find the mid - point of a segment with one endpoint on each line. For example, if \(A\) is on \(k\) and \(B\) is on \(m\) (\(k\perp m\)), the mid - point of segment \(AB\) is the center of rotation, and a \(180^{\circ}\) rotation about this mid - point maps \(k\) to \(m\).