To find the coordinates of the image of point [tex]\( H(2, -6) \)[/tex] under a [tex]\( 90^\circ \)[/tex] clockwise rotation about the origin, we follow these steps:
1. Understand the Rotation Rule:
When a point [tex]\((x, y)\)[/tex] is rotated [tex]\(90^\circ\)[/tex] clockwise about the origin, its new coordinates [tex]\((x', y')\)[/tex] are given by:
[tex]\[
(x', y') = (y, -x)
\][/tex]
2. Apply the Rotation:
Given the original point [tex]\( H(2, -6) \)[/tex]:
- [tex]\( x = 2 \)[/tex]
- [tex]\( y = -6 \)[/tex]
Substitute these values into the rotation rule:
[tex]\[
x' = y = -6
\][/tex]
[tex]\[
y' = -x = -2
\][/tex]
3. Determine the New Coordinates:
After applying the rotation, the new coordinates of [tex]\( H \)[/tex] are:
[tex]\[
H'(-6, -2)
\][/tex]
So, the image of point [tex]\( H(2, -6) \)[/tex] under a [tex]\( 90^\circ \)[/tex] clockwise rotation about the origin is:
[tex]\[
H'(-6, -2)
\][/tex]
Therefore, the correct answer is [tex]\( H'(-6, -2) \)[/tex].
