axis of symmetry of a kite geometry definition

The deltoidal icositetrahedron, deltoidal hexecontahedron, and trapezohedron are polyhedra with congruent kite-shaped facets. A kite with angles π/3, π/2, 2π/3, π/2 can also tile the plane by repeated reflection across its edges the resulting tessellation, the deltoidal trihexagonal tiling, superposes a tessellation of the plane by regular hexagons and isosceles triangles. Tilings and polyhedraĪll kites tile the plane by repeated inversion around the midpoints of their edges, as do more generally all quadrilaterals. Thus the kites are exactly the quadrilaterals that are both orthodiagonal and tangential.

The center of the incircle lies on a line of symmetry that is also a diagonal.The products of opposite sides are equal.One pair of opposite tangent lengths have equal length.The two line segments connecting opposite points of tangency have equal length.The area is one half the product of the diagonals.Ī tangential quadrilateral is a kite if and only if any one of the following conditions is true: 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, while the other circle is exterior to the kite and touches the kite on the two edges incident to the concave angle. 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. Therefore, every convex kite is a tangential quadrilateral. Additional propertiesĮvery convex kite has an inscribed circle that is, there exists a circle that is tangent to all four sides. The two interior angles of a kite that are on opposite sides of the symmetry axis are equal. 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. 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:Īlternatively, if a and b are the lengths of two unequal sides, and θ is the angle between unequal sides, then the area is 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. AreaĮvery kite is orthodiagonal, meaning that its two diagonals are at right angles to each other. Kites and isosceles trapezoids are dual: the polar figure of a kite is an isosceles trapezoid, and vice versa. If crossings are allowed, the list of quadrilaterals with axes of symmetry must be expanded to also include the antiparallelograms. 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.

The kites are the quadrilaterals that have an axis of symmetry along one of their diagonals.

Exactly one pair of opposite angles are bisected by a diagonal.

One diagonal is a line of symmetry (it divides the quadrilateral into two congruent triangles).

(In the concave case it is the extension of one of the diagonals.)

One diagonal is the perpendicular bisector of the other diagonal.

Two pairs of adjacent sides are equal (by definition).

CharacterizationsĪ quadrilateral is a kite if and only if any one of the following statements is true: The tiling that it produces by its reflections is the deltoidal trihexagonal tiling. 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 one of them is a right kite, with 60°, 90°, and 120° angles. That is, for these kites the two equal angles on opposite sides of the symmetry axis are each 90 degrees. the kites that can be inscribed in a circle) are exactly the ones formed from two congruent right triangles. The kites that are also cyclic quadrilaterals (i.e.

An equidiagonal kite that maximizes the ratio of perimeter to diameter, inscribed in a Reuleaux triangleĪmong 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.

Axis of symmetry of a kite geometry definition