Given a square with side length and parallel lines with slopes and perpendicular lines with slopes , find the area of the shaded square. Also find the ratio that makes the area of the inner square equal to 50% of the outer square.

The problem can be solved in many ways. The solution illustrated here is visual. It is done in three steps:

Step 1:

The area of the parallelogram = area of square – area of two red triangles = .

Step 2:

Cut the trapezoidal piece from the bottom of the parallelogram and attach it to the top. The parallelogram becomes a rectangle with its base on the base of the inner square. The length of the parallelogram is (according to the Pythagorean theorem) . That is the length of the shaded rectangle, and its width is to the length of the side of the inner square, .

The area of the shaded rectangle = base × height = = the area of the parallelogram.