Question

1. name the image of p(9, 1.5) after being translated p(\underline{12}, \underline{1}) 3) what is the location of the image of p(-8, 1) after a counterclockwise rotation of 90°about (-2, 6)? graph of coordinate plane with p(-8,1) marked \bigcirc (3,0) \bigcirc (3,-16) \bigcirc (-7,12) \bigcirc (9,12)

Explanation:

Step1: Translate point \( P \) to origin

Let the center of rotation be \( C(-2, 6) \). First, find the vector from \( C \) to \( P(-8, 1) \):
\( \vec{CP} = (-8 - (-2), 1 - 6) = (-6, -5) \).

Step2: Rotate the vector 90° counterclockwise

The rule for 90° counterclockwise rotation of a vector \( (x, y) \) is \( (-y, x) \).
Applying this to \( \vec{CP} = (-6, -5) \):
New vector \( \vec{CP'} = (5, -6) \).

Step3: Translate back to original center

Add the center \( C(-2, 6) \) to the rotated vector:
\( P'(x, y) = (-2 + 5, 6 + (-6)) = (3, 0) \).

Answer:

\( (3, 0) \) (corresponding to option A. \( (3, 0) \))