draw the line of reflection that reflects quadrilateral abcd onto quadrilateral abcd.

Explanation:

Step1: Find mid - points of corresponding points

For two corresponding points (P(x_1,y_1)) and (P'(x_2,y_2)) of the pre - image and image, the mid - point (M) of the line segment (PP') is given by (M(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})). For example, for point (A(4,5)) and (A'(5,1)), the mid - point (M_A) is ((\frac{4 + 5}{2},\frac{5+1}{2})=(4.5,3)). Do this for other corresponding points.

Step2: Observe the pattern of mid - points

After finding mid - points of all corresponding points (e.g., for (B(- 2,2)) and (B'(2,-2)) mid - point is ((0,0)); for (C(-6,3)) and (C'(3,-5)) mid - point is ((-1.5,-1)); for (D(-3,7)) and (D'(5,-5)) mid - point is ((1,1))), we can see that these mid - points lie on the line (y=-x).

Step3: Confirm the line of reflection

The line of reflection is the perpendicular bisector of the line segments joining corresponding points. The line (y =-x) is the perpendicular bisector of the line segments (AA'), (BB'), (CC') and (DD'). So the line of reflection is (y=-x).

Answer:

The line of reflection is (y =-x) (draw the line (y=-x) which passes through points like ((0,0)), ((1,-1)), ((-1,1)) etc. on the given coordinate grid).