1. **Problem Statement:** Determine which transformations map an equilateral triangle exactly onto itself given lines m and n and center K. 2. **Key Properties of an Equilateral Triangle:** - All sides are equal. - All angles are 60°. - The center K is the centroid, circumcenter, and incenter. - Symmetries include rotations by multiples of 120° about K and reflections across lines through vertices and midpoints. 3. **Transformations to Check:** - Clockwise rotation about K by 216° - Reflection across line n - Reflection across line m - Counterclockwise rotation about K by 240° 4. **Rotation Symmetry:** An equilateral triangle maps onto itself under rotations about K by multiples of 120°: $$120^\circ, 240^\circ, 360^\circ$$ - 216° is not a multiple of 120°, so it does not map the triangle onto itself. - 240° (counterclockwise) is a multiple of 120°, so it maps the triangle onto itself. 5. **Reflection Symmetry:** Reflections that map the triangle onto itself are those across lines of symmetry: - Lines through a vertex and the midpoint of the opposite side (like line m). - Lines through midpoints and opposite vertices (like line n). Since line m passes through a vertex and bisects the opposite side, reflection across m maps the triangle onto itself. Line n bisects each side it passes through, so it is also a line of symmetry, so reflection across n maps the triangle onto itself. 6. **Summary:** - Clockwise rotation by 216°: No - Reflection across line n: Yes - Reflection across line m: Yes - Counterclockwise rotation by 240°: Yes **Final answer:** Reflection across line n, Reflection across line m, and Counterclockwise rotation about K by 240° map the triangle exactly onto itself.