To find the coordinates of the final image after the figure undergoes a 90° counterclockwise rotation about the origin, followed by a translation of 3 units left and 8 units up, follow these steps:
1. Rotate each point 90° counterclockwise about the origin using the rotation formulas:
For a point (x, y) rotated counterclockwise by 90°:
New_x = -y
New_y = x
2. Translate each rotated point 3 units left and 8 units up.
Let's calculate it for each point:
For point R(-7, -5):
Rotating 90° counterclockwise:
New_x_R = -(-5) = 5
New_y_R = -(-7) = 7
Translating 3 units left and 8 units up:
Final_x_R = 5 - 3 = 2
Final_y_R = 7 + 8 = 15
For point S(-1, -2):
Rotating 90° counterclockwise:
New_x_S = -(-2) = 2
New_y_S = -(-1) = 1
Translating 3 units left and 8 units up:
Final_x_S = 2 - 3 = -1
Final_y_S = 1 + 8 = 9
For point T(-1, -5):
Rotating 90° counterclockwise:
New_x_T = -(-5) = 5
New_y_T = -(-1) = 1
Translating 3 units left and 8 units up:
Final_x_T = 5 - 3 = 2
Final_y_T = 1 + 8 = 9
So, the coordinates of the final image after the rotation and translation are:
- Point R: (2, 15)
- Point S: (-1, 9)
- Point T: (2, 9)