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Picture of parallelogram1/15/2024 ![]() ![]() K rect = ( B + A ) × H K_ Proof that diagonals bisect each other The area K of the parallelogram to the right (the blue area) is the total area of the rectangle less the area of the two orange triangles. The base × height area formula can also be derived using the figure to the right. The area of the parallelogram is the area of the blue region, which is the interior of the parallelogram The diagonals of a parallelogram divide it into four triangles of equal area.If two lines parallel to sides of a parallelogram are constructed concurrent to a diagonal, then the parallelograms formed on opposite sides of that diagonal are equal in area.The centers of four squares all constructed either internally or externally on the sides of a parallelogram are the vertices of a square.Unlike any other convex polygon, a parallelogram cannot be inscribed in any triangle with less than twice its area.The perimeter of a parallelogram is 2( a + b) where a and b are the lengths of adjacent sides.If it has four lines of reflectional symmetry, it is a square. If it also has exactly two lines of reflectional symmetry then it must be a rhombus or an oblong (a non-square rectangle). A parallelogram has rotational symmetry of order 2 (through 180°) (or order 4 if a square).Any non-degenerate affine transformation takes a parallelogram to another parallelogram.Any line through the midpoint of a parallelogram bisects the area.The area of a parallelogram is also equal to the magnitude of the vector cross product of two adjacent sides.The area of a parallelogram is twice the area of a triangle created by one of its diagonals.Opposite sides of a parallelogram are parallel (by definition) and so will never intersect.Thus all parallelograms have all the properties listed above, and conversely, if just one of these statements is true in a simple quadrilateral, then it is a parallelogram. There is a point X in the plane of the quadrilateral with the property that every straight line through X divides the quadrilateral into two regions of equal area.(This is an extension of Viviani's theorem.) The sum of the distances from any interior point to the sides is independent of the location of the point.The sum of the squares of the sides equals the sum of the squares of the diagonals. ![]()
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