To determine the image of the point [tex]\((5, -2)\)[/tex] under a rotation of [tex]\(90^{\circ}\)[/tex] counterclockwise about the origin, we need to follow the standard rules for rotating points in the Cartesian plane.
When we rotate a point [tex]\((x, y)\)[/tex] counterclockwise by [tex]\(90^{\circ}\)[/tex], the new coordinates [tex]\((x', y')\)[/tex] can be found using the following transformations:
[tex]\[ x' = -y \][/tex]
[tex]\[ y' = x \][/tex]
Let's apply these transformations to the given point [tex]\((5, -2)\)[/tex]:
1. Start with the given coordinates: [tex]\((5, -2)\)[/tex].
2. Apply the transformation for the 90-degree counterclockwise rotation:
[tex]\[ x' = -(-2) = 2 \][/tex]
[tex]\[ y' = 5 \][/tex]
So, after the rotation, the new coordinates of the point are [tex]\((2, 5)\)[/tex].
Now, let's check the given answer choices:
1. [tex]\((-5, -2)\)[/tex]
2. [tex]\((2, 5)\)[/tex]
3. [tex]\((-2, 5)\)[/tex]
4. [tex]\((-5, 2)\)[/tex]
From these choices, the correct coordinates [tex]\((2, 5)\)[/tex] correspond to the second option.
Therefore, the image of the point [tex]\((5, -2)\)[/tex] under a [tex]\(90^{\circ}\)[/tex] rotation counterclockwise about the origin is [tex]\((2, 5)\)[/tex], which is option 2.
