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If the quadrilateral is rectangle, then equation simplifies further since now the two diagonals are of equal length as well: 2 a 2 + 2 b 2 = 2 e 2 {\displaystyle 2a^{2}+2b^{2}=2e^{2}} Dividing by 2 yields the Euler–Pythagoras theorem:
In Euclidean plane geometry, a rectangle is a rectilinear convex polygon or a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a parallelogram containing a right angle. A rectangle with four sides of equal length is a square.
In Euclidean geometry, an equidiagonal quadrilateral is a convex quadrilateral whose two diagonals have equal length. Equidiagonal quadrilaterals were important in ancient Indian mathematics , where quadrilaterals were classified first according to whether they were equidiagonal and then into more specialized types.
The diagonals of a rhombus intersect at equal angles, while the diagonals of a rectangle are equal in length. The figure formed by joining the midpoints of the sides of a rhombus is a rectangle , and vice versa.
An equivalent condition is that the diagonals bisect each other, and are equal in length. Rectangles include squares and oblongs. Informally: "a box or oblong" (including a square). Square (regular quadrilateral): all four sides are of equal length (equilateral), and all four angles are right angles. An equivalent condition is that opposite ...
The rectangle of corollary 1 is now a symmetrical trapezium with equal diagonals and a pair of equal sides. The parallel sides differ in length by units where: = (+) It will be easier in this case to revert to the standard statement of Ptolemy's theorem:
The four sides can be split into two pairs of adjacent equal-length sides. [7] One diagonal crosses the midpoint of the other diagonal at a right angle, forming its perpendicular bisector. [9] (In the concave case, the line through one of the diagonals bisects the other.) One diagonal is a line of symmetry.
The Varignon parallelogram is a rhombus if and only if the two diagonals of the quadrilateral have equal length, that is, if the quadrilateral is an equidiagonal quadrilateral. [ 7 ] The Varignon parallelogram is a rectangle if and only if the diagonals of the quadrilateral are perpendicular , that is, if the quadrilateral is an orthodiagonal ...