To determine the new coordinates of the endpoints of the segment [tex]$\overline{PL}$[/tex] after applying the translation, follow these steps:
1. Identify the given coordinates:
- Point [tex]\(P\)[/tex] has coordinates [tex]\((4, -6)\)[/tex].
- Point [tex]\(L\)[/tex] has coordinates [tex]\((-2, 1)\)[/tex].
2. Understand the translation mapping:
- The translation specifies that we should add 5 to the x-coordinate and keep the y-coordinate the same: [tex]\((x, y) \to (x + 5, y)\)[/tex].
3. Apply the translation to point [tex]\(P\)[/tex]:
- The x-coordinate of [tex]\(P\)[/tex] is 4. After applying the translation, the new x-coordinate will be [tex]\(4 + 5 = 9\)[/tex].
- The y-coordinate of [tex]\(P\)[/tex] is [tex]\(-6\)[/tex]. Since the y-coordinate doesn't change, it remains [tex]\(-6\)[/tex].
- Therefore, the new coordinates of [tex]\(P^{\prime}\)[/tex] are [tex]\((9, -6)\)[/tex].
4. Apply the translation to point [tex]\(L\)[/tex]:
- The x-coordinate of [tex]\(L\)[/tex] is [tex]\(-2\)[/tex]. After applying the translation, the new x-coordinate will be [tex]\(-2 + 5 = 3\)[/tex].
- The y-coordinate of [tex]\(L\)[/tex] is 1. Since the y-coordinate doesn't change, it remains 1.
- Therefore, the new coordinates of [tex]\(L^{\prime}\)[/tex] are [tex]\((3, 1)\)[/tex].
So, after applying the translation, the new coordinates of the endpoints are:
- [tex]\(P^{\prime}(9, -6)\)[/tex]
- [tex]\(L^{\prime}(3, 1)\)[/tex]
Therefore, the correct option is:
[tex]\[P^{\prime}(9, -6), L^{\prime}(3, 1)\][/tex]
