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4. What is the image of [tex]\((1,-6)\)[/tex] for a [tex]\(90^{\circ}\)[/tex] counterclockwise rotation about the origin?

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To determine the image of the point [tex]\((1, -6)\)[/tex] after a [tex]\(90^{\circ}\)[/tex] counterclockwise rotation about the origin, we follow these steps:

1. Understand the transformation: A [tex]\(90^{\circ}\)[/tex] counterclockwise rotation about the origin in a Cartesian coordinate system transforms a point [tex]\((x, y)\)[/tex] to [tex]\((-y, x)\)[/tex].

2. Identify the original coordinates: The coordinates of the original point are [tex]\((1, -6)\)[/tex].

3. Apply the transformation:
- The [tex]\(x\)[/tex]-coordinate of the new point will be the negative of the original [tex]\(y\)[/tex]-coordinate. Thus, the new [tex]\(x\)[/tex]-coordinate is [tex]\(-(-6) = 6\)[/tex].
- The [tex]\(y\)[/tex]-coordinate of the new point will be the original [tex]\(x\)[/tex]-coordinate. Thus, the new [tex]\(y\)[/tex]-coordinate is [tex]\(1\)[/tex].

4. Resulting coordinates: After applying the transformation, the new coordinates are [tex]\((6, 1)\)[/tex].

Therefore, the image of the point [tex]\((1, -6)\)[/tex] after a [tex]\(90^{\circ}\)[/tex] counterclockwise rotation about the origin is [tex]\((6, 1)\)[/tex].
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