Kite (geometry)

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This article is about the geometric shape. For the flying object, see Kite.

A kite showing its equal sides and its inscribed circle.

In geometry, a kite, or deltoid, is a quadrilateral with two pairs of equal adjacent sides. Technically, the pairs of sides are disjoint congruent and adjacent. This is in contrast to a parallelogram, where the equal sides are opposite. The geometric object is named for the wind-blown, flying kite (which is itself named for a bird), which in its simple form often has this shape.

1 Properties
2 Special cases
3 There are 4 Special Kites:
4 See also
5 Notes
6 External links

[edit] Properties

The pairs of equal sides imply many properties:

One diagonal divides the kite into two isosceles triangles, and the other divides the kite into two congruent triangles
Kites always posses at least one symmetry axis; that axis is the diagonal that divides it into two congruent triangles
A kite possesses an inscribed circle; that is, there exists a circle that is tangent to all four sides.
If $d 1$ and $d 2$ are the lengths of the diagonals, then the area is

$A=\frac{d_1d_2}{2}$

Alternatively, if $a$ and $b$ are the lengths of the sides, and $θ$ the angle between unequal sides, then the area is

$A={a b \sin\theta}\,$

[edit] Special cases

When the kite is concave, it becomes an arrowhead^[1], rather than a kite, and is also the only possible concave quadrilateral.
If all the sides are the same length, the quadrilateral is called a rhombus.

[edit] There are 4 Special Kites:

Equilateral Kites: one of the "two triangles" that make up the kite has all equal sides.

m<A=m<ABD=m<ADB=60° AB≅BD≅AD

Right Kite: one of the "two triangles" that make up the kite has a 90° angle at one of it's "points"

m<C=90° DC=BC=x, DB=2x

"Y:Z" Kites: the pair of similar sides are proportionate to the other pair of similar sides

DC=BC=Yx, AB=AD=Zx Note: The only exception is 1:2 kites, which are Equilateral Right Kites

Equilateral Right Kites: a combination of both Equilateral Kites and Right Kites where one of the kite's "triangles" are equilateral and the opposite "triangle" has its "points" equal to 90°

m<A=60° m<B=m<D=105° (m<DBC=BDC=45°, m<ADB=m<ABD=60°) AB≅AD≅DB=2x, DC≅BC=x