![]() ![]() As with any crossed quadrilateral, the sum of its interior angles is 720°, allowing for internal angles to appear on the outside and exceed 180°.Ī rectangle and a crossed rectangle are quadrilaterals with the following properties in common: The interior of a crossed rectangle can have a polygon density of ☑ in each triangle, dependent upon the winding orientation as clockwise or counterclockwise.Ī crossed rectangle may be considered equiangular if right and left turns are allowed. A three-dimensional rectangular wire frame that is twisted can take the shape of a bow tie. It appears as two identical triangles with a common vertex, but the geometric intersection is not considered a vertex.Ī crossed quadrilateral is sometimes likened to a bow tie or butterfly, sometimes called an "angular eight". It has the same vertex arrangement as the rectangle. Similarly, a crossed rectangle is a crossed quadrilateral which consists of two opposite sides of a rectangle along with the two diagonals. The British flag theorem states that with vertices denoted A, B, C, and D, for any point P on the same plane of a rectangle: Crossed rectanglesĪ crossed quadrilateral (self-intersecting) consists of two opposite sides of a non-self-intersecting quadrilateral along with the two diagonals. The Japanese theorem for cyclic quadrilaterals states that the incentres of the four triangles determined by the vertices of a cyclic quadrilateral taken three at a time form a rectangle. The midpoints of the sides of any quadrilateral with perpendicular diagonals form a rectangle.Ī parallelogram with equal diagonals is a rectangle. The isoperimetric theorem for rectangles states that among all rectangles of a given perimeter, the square has the largest area. The area of a rectangle is the product of the length and width. The figure formed by joining, in order, the midpoints of the sides of a rectangle is a rhombus and vice versa.Two axes of symmetry bisect opposite angles. Two axes of symmetry bisect opposite sides. Its centre is equidistant from its sides, hence it has an incircle. Its centre is equidistant from its vertices, hence it has a circumcircle. The dual polygon of a rectangle is a rhombus, as shown in the table below. It has two lines of reflectional symmetry and rotational symmetry of order 2 (through 180°). It is isogonal or vertex-transitive: all corners lie within the same symmetry orbit. It is equiangular: all its corner angles are equal (each of 90 degrees). Star-shaped: The whole interior is visible from a single point, without crossing any edge.Ī rectangle is cyclic: all corners lie on a single circle.Simple: The boundary does not cross itself.A parallelogram is a special case of a trapezium (known as a trapezoid in North America) in which both pairs of opposite sides are parallel and equal in length.Ī trapezium is a convex quadrilateral which has at least one pair of parallel opposite sides. ![]()
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