Geometry Kite by mathedutireteam

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									                                        Geometry Kite

Geometry Kite
A kite is a quadrilateral whose four sides can be grouped into two pairs of equal-length sides that are
adjacent to each other. In contrast, a parallelogram also has two pairs of equal-length sides, but they are
opposite each other rather than adjacent. Kite quadrilaterals are named for the wind-blown, flying kites,
which often have this shape and which are in turn named for a bird. Kites are also known as deltoids,
but the word "deltoid" may also refer to a deltoid curve, an unrelated geometric object. A kite, as
defined above, may be either convex or concave, but the word "kite" is often restricted to the convex
variety. A concave kite is sometimes called a "dart" or "arrowhead", and is a type of pseudotriangle.

If all four sides of a kite have the same length (that is, if the kite is equilateral), it must be a rhombus. If
a kite is equiangular, meaning that all four of its angles are equal, then it must also be equilateral and
thus a square. The kites that are also cyclic quadrilaterals (i.e. the kites that can be inscribed in a circle)
are exactly the ones formed from two congruent right triangles. That is, for these kites the two equal
angles on opposite sides of the symmetry axis are each 90 degrees.[1] These shapes are called right
kites. An equidiagonal kite that maximizes the ratio of perimeter to diameter, inscribed in a Reuleaux
triangle Among all quadrilaterals, the shape that has the greatest ratio of its perimeter to its diameter is
an equidiagonal kite with angles π/3, 5π/12, 5π/6, 5π/12. There are only eight polygons that can tile the
plane in such a way that reflecting any tile across any one of its edges produces another tile.
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Characterizations: A quadrilateral is a kite if and only if any one of the following conditions is true:
Two disjoint pairs of adjacent sides are equal (by definition). One diagonal is the perpendicular bisector
of the other diagonal.[5] (In the concave case it is the extension of one of the diagonals.) One diagonal
is a line of symmetry (it divides the quadrilateral into two congruent triangles). One diagonal bisects a
pair of opposite angles.

Symmetry :- The kites are the quadrilaterals that have an axis of symmetry along one of their
diagonals.[7] Any non-self-crossing quadrilateral that has an axis of symmetry must be either a kite (if
the axis of symmetry is a diagonal) or an isosceles trapezoid (if the axis of symmetry passes through the
midpoints of two sides); these include as special cases the rhombus and the rectangle respectively,
which have two axes of symmetry each, and the square which is both a kite and an isosceles trapezoid
and has four axes of symmetry.[7] If crossings are allowed, the list of quadrilaterals with axes of
symmetry must be expanded to also include the antiparallelograms.

Basic properties :- Every kite is orthodiagonal, meaning that its two diagonals are at right angles to
each other. Moreover, one of the two diagonals (the symmetry axis) is the perpendicular bisector of the
other, and is also the angle bisector of the two angles it meets. One of the two diagonals of a convex
kite divides it into two isosceles triangles; the other (the axis of symmetry) divides the kite into two
congruent triangles. The two interior angles of a kite that are on opposite sides of the symmetry axis are
equal.

Area :- As is true more generally for any orthodiagonal quadrilateral, the area K of a kite may be
calculated as half the product of the lengths of the diagonals p and q. Alternatively, if a and b are the
lengths of two unequal sides, and θ is the angle between unequal sides .

Tangent circles : - Every convex kite has an inscribed circle; that is, there exists a circle that is tangent
to all four sides. Therefore, every convex kite is a tangential quadrilateral. Additionally, if a convex kite
is not a rhombus, there is another circle, outside the kite, tangent to the lines that pass through its four
sides; therefore, every convex kite that is not a rhombus is an ex-tangential quadrilateral. For every
concave kite there exist two circles tangent to all four (possibly extended) sides: one is interior to the
kite and touches the two sides opposite from the concave angle.

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