Answer:
To find the image of point P=(8,2) under a rotation of 270 degrees counterclockwise about the origin, we can use the following rotation matrix:
|cos(θ) -sin(θ)| |x| |x'| |sin(θ) cos(θ)| |y| = |y'|
where θ is the angle of rotation, x and y are the coordinates of the original point P, and x' and y' are the coordinates of the rotated point P'.
For a rotation of 270 degrees counterclockwise, θ = -270° (or θ = 90°, depending on the convention used). Thus, the rotation matrix becomes:
|cos(-270°) -sin(-270°)| |8| |x'| |sin(-270°) cos(-270°)| |2| = |y'|
Simplifying the matrix elements using the values of cosine and sine of -270 degrees, we get:
|0 1| |8| |x'| |-1 0| |2| = |y'|
Multiplying the matrices, we get:
x' = 08 + 12 = 2 y' = -18 + 02 = -8
Therefore, the image of point P=(8,2) under a rotation of 270 degrees counterclockwise about the origin is P'=(2,-8).