Draw segments XZ and YZ with equal lengths as in the picture. The three pieces
can be formed into a square as shown:
squares can be used to make a third square in this manner.
The Hint and the Answer suggest using a piece of paper
to find point Z and construct a square. (The edges of
the paper will lie on two sides of the new square. Carpenters
would use a carpenters square for finding this point.)
A different way to find point Z is to mark off the length
of a side of the small square along the bottom of the
larger square starting at the left. This length locates
point Z. Because the total length of the bottom is the
sum of the lengths of a side of each square, Z also separates
the bottom into two lengths that are the lengths of the
sides of the squares.
Draw segments XZ and YZ. The two right triangles
are the same size and shape because each has a right angle
and the two smaller sides are each a length of the original
squares. Because the triangles are the same size and shape,
XZ and YZ are the same length. They become sides of the
new square. Using the angles of the triangles along the
base, angle XZY can be shown to be a right angle.