Amelia’s mother proposes a game. “Pick two of the shapes below,” she says to Amelia. (The shapes are an equilateral triangle, a parallelogram, an isosceles trapezoid, a kite, and an ellipse. These shapes are drawn to scale.) Amelia’s mother continues: “I will draw those two shapes on a sheet of paper, in whatever position and orientation I choose, without overlapping them. Then you draw a straight line that cuts both shapes, so that each shape is divided into two congruent halves.”
Which two of the shapes should Amelia choose to guarantee that she can succeed? Given that choice of shapes, explain how Amelia can draw her line, what property of those shapes makes it possible for her to do so, and why this would not work with any other pair of these shapes.
