The Area of a Square in a Square

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.

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DETAILS

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.
Therefore: , and so .
Step 3:
The required area of the inner square is
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Related Curriculum Standards

US Common Core State Standards, Mathematics