# Solution to the Rectangle Tilted Over a Square Puzzle +-- {.image} [[RectangleTiltedOveraSquare.png:pic]] > An orange rectangle and a black square. What’s the shaded area? =-- ## Solution by [[Similar Triangles]] +-- {.image} [[RectangleTiltedOveraSquareLabelled.png:pic]] =-- In the above diagram, triangles $A B C$ and $C D E$ are [[similar]] since they are [[right-angled triangles]] and angles $A \hat{C} B$ and $D \hat{C} E$ add together to give $90^\circ$. Therefore the ratio of lengths $A C : B C$ and $C E : D E$ are equal. Since $E D$ is twice $B C$, $E C$ is therefore twice $C A$, and hence is of length $8$. The shaded area is therefore $4 \times 8 = 32$. ## Solution by [[Invariance Principle]] The shaded rectangle can be tilted by varying amounts. Drawing it vertically, so that points $A$ and $B$ coincide, means that the outer square has side length $8$ and the height of the rectangle is the same as the side length.