Kite- A Quadrilateral

Quadrilaterals – Kite Properties:

A polygon is a plane figure which is bounded by finite line segments to form a closed figure. A 4-sided polygon is known as a quadrilateral.

The sum of interior angles of a quadrilateral is always 360°. The number of diagonals in a n-sided polygon is given by \( \frac {n(n-3)}{2} \) . Thus, a quadrilateral has 2 diagonals. Also, for any n-sided polygon, the sum of exterior angles of any polygon is always 360°.

Kite is a quadrilateral in which two pairs of adjacent sides are of equal length and the diagonals intersect each other at right angles. The figure shown below represents a kite.

Quadrilateral

Figure 1 A Kite

The fig 1 above represents a quadrilateral and it can be represented as:

Quadrilateral

Figure 2

In quadrilateral ABCD, the sides AB = BC and AD = CD. Thus, as the adjacent pair of sides is equal in length, therefore ABCD is a special type of quadrilateral known as a kite.

Kite Properties:

  • i)  Diagonals intersect at right angles.

    In a kite, the diagonals intersect at right angles. This can be proved as follows.

    From fig. 2 above, in ∆ABD and ∆BCD

AB=BC

AD=CD

BD is common.

Therefore ∆ABD ≅ ∆BCD (SSS rule of congruency)

Also, in ∆ABC and ∆ADC

AB = BC and AD = CD

Thus, ∆ABC is an isosceles triangle.

⇒ ∠BAO = ∠BCO

Thus, ∆ABO ≅ ∆BCO (ASA rule of congruency)

⇒∠AOB = ∠BOC (CPCT)

Also, ∠AOB + ∠BOC = 180° (Linear pair)

⇒ ∠AOB = ∠BOC = 90°

(NOTE: CPCT stands for congruent parts of congruent triangle)

  •  ii)  One of the diagonals is the perpendicular bisector of another.

    As seen above ∆ABO ≅ ∆BCO and ∠AOB = ∠BOC = 90°

    ⇒ AO = OC (CPCT)

    Thus BD bisects AC at right angles.

  • iii)  Angles between unequal sides are equal.

    Since, ∆BAD ≅ ∆BCD

    ⇒ ∠BAD = ∠BCD (CPCT)

  • iv)  Area of a kite:

    If length of both diagonals is given as p and q as shown in the figure given below, area (A) of kite can be given as:

    Quadrilateral

    Figure 3 Area of Kite

A = \( \frac {p \times q}{2} \)

  • v) Rhombus as a special case of kite:

If all the sides of a kite become equal in length then that kite becomes a rhombus as sides are of equal length and diagonals are perpendicular to each other.

Click to read about Parallelogram Properties and more.


Practise This Question

In which of the following situations, the heavier of the two particles has smaller de Broglie wavelength? The two particles