To find the image of the point \((-2, -7)\) after a [tex]$90^\circ$[/tex] counterclockwise rotation about the origin, follow these steps:
1. Understand the Rotation Transformation:
In a rotation of [tex]$90^\circ$[/tex] counterclockwise around the origin, every point \((x, y)\) is transformed to the point \((-y, x)\).
2. Apply the Transformation:
- For the given point \((-2, -7)\), identify \(x\) and \(y\):
- \(x = -2\)
- \(y = -7\)
- Using the rotation rule \((-y, x)\):
- The new \(x\)-coordinate will be the negative of the original \(y\)-coordinate: \(-(-7) = 7\).
- The new \(y\)-coordinate will be the original \(x\)-coordinate: \(-2\).
3. Write the Coordinates of the Image:
- The new coordinates after the transformation are \((7, -2)\).
Therefore, the image of the point [tex]\((-2, -7)\)[/tex] after a [tex]\(90^\circ\)[/tex] counterclockwise rotation about the origin is [tex]\((7, -2)\)[/tex].
