Question

1. draw a line passing through point p that is parallel to line l. draw a second line passing through point p and that is distinct (i.e., different) from the first one. what do you notice? 2. translate line l along the vector $overrightarrow{ab}$. what do you notice about l and its image, l?

Explanation:

Step1: Recall parallel - line property

In a plane, through a point not on a given line, there is exactly one line parallel to the given line. So, it is impossible to draw two distinct lines passing through point \(P\) that are parallel to line \(L\) in a plane.

Step2: Recall translation property

When a line \(L\) is translated along a vector \(\overrightarrow{AB}\), the original line \(L\) and its image \(L'\) are parallel. In the case where the vector \(\overrightarrow{AB}\) lies on the line \(L\), the original line \(L\) and its image \(L'\) coincide.

Answer:

1. It is impossible to draw two distinct lines passing through point \(P\) that are parallel to line \(L\) in a plane as there is exactly one line parallel to a given line through a point not on the line in a plane.
2. If the vector \(\overrightarrow{AB}\) lies on line \(L\), \(L = L'\); in general, \(L\parallel L'\) when translating line \(L\) along vector \(\overrightarrow{AB}\).