To determine the correct set of image points for triangle [tex]\(X'Y'Z'\)[/tex] after rotating triangle [tex]\(XYZ\)[/tex] [tex]\(90^\circ\)[/tex] clockwise about the origin, let's follow these steps:
Step 1: Understanding the 90° Clockwise Rotation
When a point [tex]\((x, y)\)[/tex] is rotated [tex]\(90^\circ\)[/tex] clockwise around the origin, it is transformed into the point [tex]\((y, -x)\)[/tex].
Step 2: Apply the Rotation to Each Vertex
- Vertex [tex]\(X(-1, -3)\)[/tex]:
[tex]\[
(x, y) = (-1, -3) \implies (y, -x) = (-3, 1)
\][/tex]
So, [tex]\(X' = (-3, 1)\)[/tex].
- Vertex [tex]\(Y(0, 0)\)[/tex]:
[tex]\[
(x, y) = (0, 0) \implies (y, -x) = (0, 0)
\][/tex]
So, [tex]\(Y' = (0, 0)\)[/tex].
- Vertex [tex]\(Z(1, -3)\)[/tex]:
[tex]\[
(x, y) = (1, -3) \implies (y, -x) = (-3, -1)
\][/tex]
So, [tex]\(Z' = (-3, -1)\)[/tex].
Step 3: Collate the Image Points
After the rotation, the image points [tex]\(X', Y', Z'\)[/tex] form the set:
[tex]\[
X'(-3, 1), Y'(0, 0), Z'(-3, -1)
\][/tex]
This matches with one of the given options:
[tex]\[
\boxed{X(-3, 1), Y(0, 0), Z(-3, -1)}
\][/tex]
