A9. A unit square has an equilateral triangle drawn inside it, with a common edge. Four of these squares are placed together to make a larger square. Four vertices of the triangles are joined up to form a square, which is shaded and shown in the diagram. What is the area of the shaded square?
Solution
2

First note that the side-length of the equilateral triangle is equal to the side-length of the unit square, that is 1 . So the diagonals of the shaded square have length 2 .
Let the side-length of the shaded square be $l$.
Then, by Pythagoras' Theorem, $l^{2}+l^{2}=2^{2}$. So $l^{2}=2$.
Hence the area of the shaded square is 2 square units.