1. **Problem Statement:** Rotate each given line by the specified angle around the origin and find the new coordinates of points on those lines. 2. **Rotation Formulas:** - Rotation by 180° around the origin: $$ (x,y) \to (-x,-y) $$ - Rotation by 90° counterclockwise: $$ (x,y) \to (-y,x) $$ - Rotation by 90° clockwise (or 270° counterclockwise): $$ (x,y) \to (y,-x) $$ 3. **Apply rotations to the example point (1,8) for line 1:** - Rotate 180°: $$ (1,8) \to (-1,-8) $$ (given) 4. **Line 1, 2, 3 rotations (all 180°):** - For any point $(x,y)$ on these lines, new point is $$ (-x,-y) $$ 5. **Line 4 rotation (270° clockwise):** - Using formula for 270° clockwise rotation: $$ (x,y) \to (y,-x) $$ 6. **Line 5 rotation (90° counterclockwise):** - Using formula for 90° counterclockwise rotation: $$ (x,y) \to (-y,x) $$ 7. **Line 6 rotation (90° clockwise):** - Using formula for 90° clockwise rotation: $$ (x,y) \to (y,-x) $$ 8. **Summary:** - Lines 1, 2, 3: rotate 180° $$ (x,y) \to (-x,-y) $$ - Line 4: rotate 270° clockwise $$ (x,y) \to (y,-x) $$ - Line 5: rotate 90° counterclockwise $$ (x,y) \to (-y,x) $$ - Line 6: rotate 90° clockwise $$ (x,y) \to (y,-x) $$ This completes the rotation transformations for each line as instructed.