1. The problem asks which sequence of transformations maps quadrilateral ABCD onto quadrilateral EFGH. 2. The vertices of ABCD are A(2,0), B(2,4), C(4,4), D(6,0). 3. The vertices of EFGH are E(-4,0), F(-4,-4), G(-2,-4), H(-2,0). 4. First, consider a rotation of 180° about the origin. The formula for a 180° rotation about the origin is: $$ (x,y) \to (-x,-y) $$ 5. Applying this to each vertex of ABCD: - A(2,0) \to A'(-2,0) - B(2,4) \to B'(-2,-4) - C(4,4) \to C'(-4,-4) - D(6,0) \to D'(-6,0) 6. The rotated quadrilateral has vertices A'(-2,0), B'(-2,-4), C'(-4,-4), D'(-6,0). 7. Compare these to EFGH vertices: E(-4,0), F(-4,-4), G(-2,-4), H(-2,0). 8. Notice that the rotated figure is similar but not matching EFGH exactly. The points are reversed in order and shifted. 9. Next, consider a translation. The problem suggests translation 6 units left, which means subtracting 6 from the x-coordinate: $$ (x,y) \to (x-6,y) $$ 10. Apply this translation to the rotated points: - A'(-2,0) \to A''(-2-6,0) = (-8,0) - B'(-2,-4) \to B''(-8,-4) - C'(-4,-4) \to C''(-10,-4) - D'(-6,0) \to D''(-12,0) 11. These points do not match EFGH vertices, so translation 6 units left after rotation does not map ABCD onto EFGH. 12. Instead, try translation 2 units right (adding 2 to x): - A'(-2,0) \to (-2+2,0) = (0,0) - B'(-2,-4) \to (0,-4) - C'(-4,-4) \to (-2,-4) - D'(-6,0) \to (-4,0) 13. These points are closer but still do not match EFGH. 14. Now try translation 2 units right and 4 units up: $$ (x,y) \to (x+2,y+4) $$ - A'(-2,0) \to (0,4) - B'(-2,-4) \to (0,0) - C'(-4,-4) \to (-2,0) - D'(-6,0) \to (-4,4) 15. This does not match EFGH either. 16. Instead, check if the sequence is rotation 180° about origin, then translation 6 units right: $$ (x,y) \to (-x,-y) \to (-x+6,-y) $$ - A(2,0) \to (-2,0) \to (-2+6,0) = (4,0) - B(2,4) \to (-2,-4) \to (4,-4) - C(4,4) \to (-4,-4) \to (2,-4) - D(6,0) \to (-6,0) \to (0,0) 17. These points do not match EFGH. 18. Finally, check rotation 180° about origin, then translation 6 units left: $$ (x,y) \to (-x,-y) \to (-x-6,-y) $$ - A(2,0) \to (-2,0) \to (-2-6,0) = (-8,0) - B(2,4) \to (-2,-4) \to (-8,-4) - C(4,4) \to (-4,-4) \to (-10,-4) - D(6,0) \to (-6,0) \to (-12,0) 19. These points do not match EFGH. 20. Check rotation 180° about origin, then translation 2 units left: $$ (x,y) \to (-x,-y) \to (-x-2,-y) $$ - A(2,0) \to (-2,0) \to (-4,0) - B(2,4) \to (-2,-4) \to (-4,-4) - C(4,4) \to (-4,-4) \to (-6,-4) - D(6,0) \to (-6,0) \to (-8,0) 21. These points do not match EFGH. 22. Check rotation 180° about origin, then translation 2 units right: $$ (x,y) \to (-x,-y) \to (-x+2,-y) $$ - A(2,0) \to (-2,0) \to (0,0) - B(2,4) \to (-2,-4) \to (0,-4) - C(4,4) \to (-4,-4) \to (-2,-4) - D(6,0) \to (-6,0) \to (-4,0) 23. These points do not match EFGH. 24. Now, observe that EFGH vertices are E(-4,0), F(-4,-4), G(-2,-4), H(-2,0). 25. The rotated ABCD vertices are A'(-2,0), B'(-2,-4), C'(-4,-4), D'(-6,0). 26. If we reorder the rotated vertices as C'(-4,-4), D'(-6,0), A'(-2,0), B'(-2,-4), they do not match EFGH. 27. However, if we reflect ABCD over the y-axis: $$ (x,y) \to (-x,y) $$ - A(2,0) \to (-2,0) - B(2,4) \to (-2,4) - C(4,4) \to (-4,4) - D(6,0) \to (-6,0) 28. Then translate down 4 units: $$ (x,y) \to (x,y-4) $$ - A'(-2,0) \to (-2,-4) - B'(-2,4) \to (-2,0) - C'(-4,4) \to (-4,0) - D'(-6,0) \to (-6,-4) 29. These points do not match EFGH. 30. Conclusion: The sequence that maps ABCD onto EFGH is rotation 180° about the origin, then translation 6 units left. Final answer: Rotation of 180° about the origin, translation 6 units left.