To find the coordinates of the vertices of the pre-image of a triangle that has been rotated [tex]$90^{\circ}$[/tex] about the origin, we need to reverse the rotation. Here's the step-by-step explanation:
1. Understand the Geometry of Rotation:
- A [tex]$90^{\circ}$[/tex] rotation about the origin changes each point [tex]$(x, y)$[/tex] to [tex]$(-y, x)$[/tex]. Conversely, to reverse a [tex]$90^{\circ}$[/tex] rotation, we change each point [tex]$(x, y)$[/tex] to [tex]$(y, -x)$[/tex].
2. Given Coordinates of the Image:
- The image vertices after rotation are given as [tex]\(A' (6, 3)\)[/tex], [tex]\(B' (-2, 1)\)[/tex], and [tex]\(C' (1, 7)\)[/tex].
3. Finding the Pre-Image Coordinates:
- For [tex]\(A'\)[/tex]: The coordinates are [tex]\( (6, 3) \)[/tex].
- To reverse the rotation: [tex]\((6, 3) \rightarrow (3, -6)\)[/tex].
- Thus, [tex]\(A\)[/tex] is [tex]\((3, -6)\)[/tex].
- For [tex]\(B'\)[/tex]: The coordinates are [tex]\((-2, 1)\)[/tex].
- To reverse the rotation: [tex]\((-2, 1) \rightarrow (1, 2)\)[/tex].
- Thus, [tex]\(B\)[/tex] is [tex]\((1, 2)\)[/tex].
- For [tex]\(C'\)[/tex]: The coordinates are [tex]\((1, 7)\)[/tex].
- To reverse the rotation: [tex]\((1, 7) \rightarrow (7, -1)\)[/tex].
- Thus, [tex]\(C\)[/tex] is [tex]\((7, -1)\)[/tex].
4. Summary:
- The coordinates of the vertices of the pre-image are:
- [tex]\( A (3, -6) \)[/tex]
- [tex]\( B (1, 2) \)[/tex]
- [tex]\( C (7, -1) \)[/tex]
Hence, the coordinates of the vertices of the pre-image are:
[tex]\( \boxed{ (3, -6), (1, 2), (7, -1) } \)[/tex].
