Parallelogram

Parallelogram

A parallelogram ( other Greek παραλληληγραμμον from παράλληλος is a parallel line and γραμμή is a line) is a quadrilateral in which the opposite sides are pairwise parallel, that is, they lie on parallel lines. Particular cases of a parallelogram are a rectangle , a square and a rhombus .

Content

1 Properties
2 Signs of a parallelogram
3 Area of parallelogram
4 See also
5 notes

Properties

The opposite sides of the parallelogram are equal, and the diagonals at the intersection point are divided in half.

Opposite angles of the parallelogram are equal, and the sum of the adjacent ones is 180 °.

Opposite sides of the parallelogram are equal.
The opposite angles of the parallelogram are equal.
The sum of the angles adjacent to one side is 180 ° (by the property of parallel lines).
The parallelogram diagonals intersect, and the intersection point divides them in half:
$\left|AO\right|=\left|OC\right|,\left|BO\right|=\left|OD\right|$ ${\ displaystyle \ left | AO \ right | = \ left | OC \ right |, \ left | BO \ right | = \ left | OD \ right |}$ .
The intersection point of the diagonals is the center of symmetry of the parallelogram.
A parallelogram diagonal is divided into two equal triangles.
The middle lines of the parallelogram intersect at the intersection of its diagonals. At this point, its two diagonals and its two middle lines are divided in half.
Parallelogram identity : the sum of the squares of the diagonals of a parallelogram is equal to twice the sum of the squares of its two adjacent sides: let a be the length of side AB , b be the length of side BC $d_{1}$ ${\ displaystyle d_ {1}}$ and $d_{2}$ ${\ displaystyle d_ {2}}$ - the length of the diagonals; then
$d_{1}^{2}+d_{2}^{2}=2(a^{2}+b^{2}).$ ${\ displaystyle d_ {1} ^ {2} + d_ {2} ^ {2} = 2 (a ^ {2} + b ^ {2}).}$

The parallelogram identity is a simple consequence of the Euler formula for an arbitrary quadrilateral : the quadrupled square of the distance between the midpoints of the diagonals is the sum of the squares of the sides of the quadrilateral minus the sum of the squares of its diagonals . In a parallelogram, the opposite sides are equal, and the distance between the midpoints of the diagonals is zero.

An affine transformation always translates a parallelogram into a parallelogram. For any parallelogram, there is an affine transformation that maps it to a square.

Signs of a parallelogram

Quadrilateral ABCD is a parallelogram if one of the following conditions is satisfied (in this case, all the others are also fulfilled):

In a quadrilateral without self-intersections, two opposite sides are simultaneously equal and parallel: $AB=CD,AB\parallel CD$ ${\ displaystyle AB = CD, AB \ parallel CD}$ .
All opposite angles are equal in pairs: $\angle A=\angle C,\angle B=\angle D$ ${\ displaystyle \ angle A = \ angle C, \ angle B = \ angle D}$ .
In a quadrilateral without self-intersections, all opposite sides are equal in pairs: $AB=CD,BC=DA$ ${\ displaystyle AB = CD, BC = DA}$ .
All opposite sides are parallel in pairs: $AB\parallel CD,BC\parallel DA$ ${\ displaystyle AB \ parallel CD, BC \ parallel DA}$ .
Diagonals are divided at the point of their intersection in half: $AO=OC,BO=OD$ ${\ displaystyle AO = OC, BO = OD}$ .
The sum of adjacent angles is 180 degrees: $\angle A+\angle B=180^{\circ },\angle B+\angle C=180^{\circ },\angle C+\angle D=180^{\circ },\angle D+\angle A=180^{\circ }$ ${\ displaystyle \ angle A + \ angle B = 180 ^ {\ circ}, \ angle B + \ angle C = 180 ^ {\ circ}, \ angle C + \ angle D = 180 ^ {\ circ}, \ angle D + \ angle A = 180 ^ {\ circ}}$ .
The sum of the distances between the midpoints of the opposite sides of a convex quadrangle is equal to its half-perimeter.
The sum of the squares of the diagonals is equal to the sum of the squares of the sides of the convex quadrilateral: $AC^{2}+BD^{2}=AB^{2}+BC^{2}+CD^{2}+DA^{2}$ ${\ displaystyle AC ^ {2} + BD ^ {2} = AB ^ {2} + BC ^ {2} + CD ^ {2} + DA ^ {2}}$ .

Parallelogram Area

Here are formulas that are specific to a parallelogram. See also formulas for the area of arbitrary quadrangles .

The area of the parallelogram is equal to the product of its base by height:

S=ah

{\ displaystyle S = ah}

where

a

{\ displaystyle a}

- side

h

{\ displaystyle h}

- the height drawn to this side.

The area of the parallelogram is equal to the product of its sides by the sine of the angle between them:

S=ab\sin \alpha ,

{\ displaystyle S = ab \ sin \ alpha,}

Where

a

{\ displaystyle a}

and

b

{\ displaystyle b}

- the parties, and

\alpha

{\ displaystyle \ alpha}

- angle between the sides

a

{\ displaystyle a}

and

b

{\ displaystyle b}

.

The parallelogram area can also be expressed through the sides $a,\ b$ ${\ displaystyle a, \ b}$ and the length of any of the diagonals $d$ ${\ displaystyle d}$ by Heron’s formula as the sum of the areas of two equal adjacent triangles:

S=2\cdot {\sqrt {p(p-a)(p-b)(p-d)}}

{\ displaystyle S = 2 \ cdot {\ sqrt {p (pa) (pb) (pd)}}}

Where

p=(a+b+d)/2.

{\ displaystyle p = (a + b + d) / 2.}

Parallelogram

Content

Properties

Signs of a parallelogram

Parallelogram Area

See also

Notes

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