Sunday, June 22, 2008

Prallelogram


Parallelogram
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Parallelogram

A parallelogram is a quadrilateral with opposite sides parallel (and therefore opposite angles equal). A quadrilateral with equal sides is called a rhombus, and a parallelogram whose angles are all right angles is called a rectangle. And, since a square is a degenerate case of a rectangle, both squares and rectangles are special types of parallelograms.

The polygon diagonals of a parallelogram bisect each other (Casey 1888, p. 2).

The angles of a parallelogram satisfy the identities

A=C
(1)
B=D
(2)

and

 A+B=180 degrees.
(3)

A parallelogram of base b and height h has area

 A=bh=absinA=absinB.
(4)

The height of a parallelogram is

 h=asinA=asinB,
(5)

and the polygon diagonals p and q are

p=sqrt(a^2+b^2-2abcosA)
(6)
=sqrt(a^2+b^2+2abcosB)
(7)
q=sqrt(a^2+b^2+2abcosA)
(8)
=sqrt(a^2+b^2-2abcosB)
(9)

(Beyer 1987).

The sides a, b, c, d and diagonals p, q of a parallelogram satisfy

 p^2+q^2=2(a^2+b^2)
(10)

(Casey 1888, p. 22).

The area of the parallelogram with sides formed by the vectors u=(u_x,u_y) and v=(v_x,v_y) is

A=uxv
(11)
=det(uv)
(12)
=u_xv_y-u_yv_x,
(13)

where uxv is the two-dimensional cross product and detA is the determinant.

ParallelogramTheorem

As shown by Euclid, if lines parallel to the sides are drawn through any point on a diagonal of a parallelogram, then the parallelograms not containing segments of that diagonal are equal in area (and conversely), so in the above figure, A_1=A_2 (Johnson 1929).

ParallelogramSquares

The centers of four squares erected either internally or externally on the sides of a parallelograms are the vertices of a square (Yaglom 1962, pp. 96-97; Coxeter and Greitzer 1967, p. 84).

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