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What is a Kite in Geometry – (Definition, Shape, Properties & Examples)

what is a kite in geometry

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There are many types of quadrilaterals in geometry. Some of these are parallelograms such as square, rhombus, and rectangle and some are not parallelograms such as trapezium and kite. The most common example of a kite shape is a flying kite. A kite is also called a ‘deltoid’ which means ‘having a triangular shape’.

Let’s learn what is a kite in geometry, its shape, and its properties with examples and proofs.

What is a Kite in Geometry?

A kite is a quadrilateral in which four sides can be grouped into two pairs of equal-length sides that are adjacent to each other and the diagonals intersect each other at right angles. In other words, in a kite two pairs of adjacent sides are equal.

what is a kite in geometry

In the above figure $\text{ABCD}$ is a kite, where, the adjacent sides $\text{AB}$ and $\text{DA}$ are equal and similarly, the adjacent sides $\text{BC}$ and $\text{CD}$ are equal. The diagonals $\text{AC}$ and $\text{BD}$ are perpendicular to each other. The longer diagonal $\text{AC}$ is called the major diagonal and the shorter diagonal $\text{BD}$ is called the minor diagonal. The major diagonal bisects the minor diagonal.

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Properties of Kite

As learned above in a kite two pairs of adjacent sides are equal and the diagonals are perpendicular to each other and unequal. Let’s now look at some of the important properties of a kite.

what is a kite in geometry

The following are the most important properties of a kite.

  • A kite has two pairs of adjacent equal sides. In the above figure, $\text{AB} = \text{DA}$ and $\text{BC} = \text{CD}$
  • Since it’s a quadrilateral the sum of the interior angles of a kite is equal to $360^{\circ}$
  • In a kite one pair of opposite angles (obtuse) are equal. In the above figure, $\angle \text{B} = \angle \text{D}$
  • In a kite, the minor diagonal is bisected by the major diagonal.  In the above figure, the diagonal $\text{BD}$ is bisected by the diagonal $\text{AC}$, i.e., $\text{BO} = \text{OD}$
  • The diagonals are perpendicular to each other. In the above figure, $\text{AC} \perp \text{BD}$

Diagonals of a Kite

As learned above, there are two diagonals in a kite – the major diagonal and the minor diagonal. Now, let’s learn about the properties of the diagonals of a kite.

what is a kite in geometry

The following are the most important properties of the diagonals of a kite.

  • The two diagonals of a kite are of different lengths – the longer diagonal is called the major diagonal and the shorter diagonal is called the minor diagonal
  • A pair of diagonally opposite angles of a kite are said to be congruent
  • The major diagonal(longer diagonal) bisects the minor diagonal(shorter diagonal). In the above figure, diagonal $\text{AC}$ bisects diagonal $\text{BD}$  
  • The major diagonal(longer diagonal) bisects the pair of opposite angles. In the above figure, $\angle \text{CAB} = \angle \text{DAC}$, and $\angle \text{BCA} = \angle \text{ACD}$
  • The minor diagonal of a kite forms two isosceles triangles. In the above figure, diagonal $\text{BD}$ forms two isosceles triangles – $\triangle \text{ABD}$ and $\triangle \text{BCD}$.
    • In $\triangle \text{ABD}$,  the sides $\text{AB}$ and $\text{DA}$ are equal 
    • In $\triangle \text{BCD}$,  the sides $\text{BC}$ and $\text{CD}$ are equal 
  • The longer diagonal forms two congruent triangles. In the above figure, diagonal $\text{AC}$ forms two congruent triangles – $\triangle \text{ABC}$ and $\triangle \text{ADC}$

In a Kite One Pair of Opposite Angles (Obtuse) are Equal

One of the properties of a kite is that a pair of opposite obtuse angles are equal.

what is a kite in geometry

Consider the above figure. In a kite $\text{ABCD}$, $\angle \text{ABC} = \angle \text{ADC}$. Let’s prove the statement.

In $\triangle \text{ABD}$, $\angle \text{ABD} = \angle \text{ADB}$ ($\triangle \text{ABD}$ is an isosceles triangle, with $\text{AB} = \text{AD}$) ——————– (1)

In $\triangle \text{CBD}$, $\angle \text{CBD} = \angle \text{CDB}$ ($\triangle \text{CBD}$ is an isosceles triangle, with $\text{BC} = \text{CD}$) ——————– (2)

Adding (1) and (2), we get $\angle \text{ABD} + \angle \text{CBD} = \angle \text{ADB} + \angle \text{CDB}$

$=> \angle \text{ABC} = \angle \text{ADC}$

Axiom, Postulate & Theorem

The Longer Diagonal Forms Two Congruent Triangles

Another important property of a kite is that the longer diagonal forms two congruent triangles.

what is a kite in geometry

Consider the above figure. In a kite $\text{ABCD}$, $\triangle \text{ABC} \cong \triangle \text{ADC}$. Let’s prove the statement.

In $\triangle \text{ABC}$ and $\triangle \text{ADC}$

$\text{AB} = \text{AD}$ (Adjacent sides of a kite)

$\text{BC} = \text{CD}$ (Adjacent sides of a kite)

$\text{AC} = \text{AC}$ (Common Side)

Therefore, $\triangle \text{ABC} \cong \triangle \text{ADC}$ (SSS Congruence Criterion)

Diagonals of the Kite are Perpendicular to Each Other

In a kite, the diagonals are perpendicular to each other.

what is a kite in geometry

In the above figure, $\text{ABCD}$ is a kite and $\text{AC}$, $\text{BD}$ are its diagonals. The diagonals are perpendicular to each other, i.e., $\text{AC} \perp \text{BD}$.

Let’s see how to prove the above statement.

In $\triangle \text{ABC}$ and $\triangle \text{ADC}$

$\text{AB} = \text{AD}$ (Adjacent sides of a kite)

$\text{BC} = \text{CD}$ (Adjacent sides of a kite)

$\text{AC} = \text{AC}$ (Common side)

$=> \triangle \text{ABC} \cong \triangle \text{ADC}$ (SSS Congruence Criterion)

Therefore, $\angle \text{CAB} = \angle \text{CAD}$ (Corresponding Parts of Congruent Triangles)

$=> \angle \text{OAB} = \angle \text{OAD}$ ———————————– (1)

In $\triangle \text{ABO}$ and $\triangle \text{ADO}$

$\angle \text{OAB} = \angle \text{OAD}$  (From (1))

$\text{AB} = \text{AD}$ (Adjacent Sides of a Kite)

$\text{AO} = \text{AO}$ (Common Side)

$=> \triangle \text{ABO} \cong \triangle \text{ADO}$ (SAS Congruence Criterion)

Therefore, $\angle \text{AOB} = \angle \text{AOD}$ (Corresponding Parts of Congruent Triangles) ———— (2)

But $\text{BOD}$ is a straight line, therefore, $\angle \text{BOD} = 180^{\circ}$

$=>\angle \text{AOB} + \angle \text{AOD} = 180^{\circ}$

$=>\angle \text{AOB} + \angle \text{AOB} = 180^{\circ}$ (From (2))

$=>2\angle \text{AOB} = 180^{\circ}$ 

$=>\angle \text{AOB} = 90^{\circ}$ 

Thus, $\text{AC} \perp \text{BD}$

Practice Problems

  1. What is a kite?
  2. State True or False
    1. All sides of a kite are equal
    2. The diagonals of a kite are equal
    3. The diagonals of a kite are unequal
    4. Two pairs of adjacent sides of a kite are equal
    5. Two pairs of opposite sides of a kite are equal
    6. One pair of adjacent angles of a kite are equal
    7. One pair of opposite angles of a kite are equal
    8. The major diagonal bisects the angles it connects
    9. The minor diagonal bisects the angles it connects
    10. The diagonals of a kite are parallel
    11. The diagonals of a kite are perpendicular

FAQs

What is a kite in Geometry?

A kite is a quadrilateral in which four sides can be grouped into two pairs of equal-length sides that are adjacent to each other and the diagonals intersect each other at right angles.

Are the diagonals of a kite equal?

No, the diagonals of a kite are not equal. The longer diagonal is called the major diagonal and the shorter diagonal is called the minor diagonal.

Are the diagonals of a kite perpendicular?

Yes, the diagonals of a kite are perpendicular to each other.

Do the diagonals of a kite bisect each other?

No, both diagonals do not bisect each other. Only, the major diagonal(longer diagonal) bisects the minor diagonal(shorter diagonal).

Conclusion

A kite is a quadrilateral in which four sides can be grouped into two pairs of equal-length sides that are adjacent to each other. There are two unequal diagonals in a kite intersecting at right angles such that the longer diagonal bisects the shorter diagonal.

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