1. A square and a square can be cut into pieces that will fit together to form a third square.
a). Find the length of a side of the third square.
b).In the diagram at right, mark P on segment DC so that PD = 3, then draw segments PA and PF. Calculate the lengths of these segments.
c). Segments PA and PF divide the squares into pieces. Arrange the pieces to form the third square.
2. (Continuation) Change the sizes of the squares to AD = 8 and EF = 4, and redraw the diagram. Where should point P be marked this time? Form the third square again.
3.(Continuation) Will the preceding method always produce pieces that form a new square? If your answer is yes, prepare a written explanation. If your answer is no, provide a counterexample—two specific squares that cannot be converted to a single square.
