Kite - Quadrilaterals

A Kite is a special type of quadrilateral that is easily recognizable by its unique shape, resembling the traditional toy flown on a string. In geometry, a kite has two pairs of adjacent sides that are of equal length. This distinctive feature sets it apart from other quadrilaterals like squares, rectangles, and parallelograms.

Diagonals of kite intersect each other at right angles. It is one of the unique quadrilateral and has some interesting properties that are covered below in the article. In this article, we will learn about, Kite Quadrilateral, Properties of kites, Examples, and others, in detail.

What is a Kite?

Kite is a quadrilateral, i.e. it is a polygon with four sides. In a kite, we have 2 pairs of equal-length sides and they are adjacent to each other. The image of a Kite is added below,

👁 Kite

Diagonals of a Kite

A kite have two diagonals, and the properties of the diagonls of the kite are added below,

• Diagonals of the kite are not equal.
• Digonal of kite are perpendicular to each other, they intersect each other right angles.
• Shorter diagonal of a kite forms two Iscosceles Triangles.
• Longer diagonal of a kite forms two congruent triangle by SSS property of Congruence.

Angles in a Kite

In a kite we have four angles, as it a quadrilateral. The properties of the interior angles of the kite are,

• The sum of all the angle of the kite is 360°.
• Any one pair of angles in kite (obtuse angle pair) are equal.

Properties of Kite

Various properties of the kite are added below,

• Kite has 2 diagonals that intersect each other at right angles.
• A kite is symmetrical about its main diagonal.
• Angles opposite to the main diagonal are equal.
• The kite can be viewed as a pair of congruent triangles with a common base.
• The shorter diagonal divides the kite into 2 isosceles triangles.
• The area of the kite is 1/2 × d₁ × d₂

Theorem: Diagonals of Kite Intersect at Right Angles

The interesting properties of the kite is that its diagonal are always perpendicular to each other. This is proved below, we have a kite ABCD, whose diagonal intersect each other at point O.

In ∆ABD and ∆BCD

AB = BC (Property of Kite)

AD = CD (Property of Kite)

BD = BD (Common Side)

Thus, ∆ABD ≅ ∆BCD (SSS congruency)

Now, in ∆ABC and ∆ADC

AB = BC (Property of Kite)

Hence  ∆ABC is an isosceles triangle.

AD = CD (Property of Kite)

Hence ∆ADC is an isosceles triangle.

∠BAO = ∠BCO

BO = BO (Common Side)

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

Now we know ∠AOB = ∠BOC

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

Hence, ∠AOB = ∠BOC = 90°

Hence diagonals of kite intersect at right angles.

Formulas for Kite

There are two common formulas related to kite i.e.,

• Area of Kite
• Perimeter of Kite

Area of Kite

Area of Kite is calculated by the formula for area of quadrilateral. Generally Area of Kite is calculated by the formula added below,

Area of Kite = 1/2 × d₁ × d₂

Where,

• d₁ is Shorter Diagonal of Kite
• d₂ is Longer Diagonal of Kite

Perimeter of Kite

Perimeter of Kite is calculated by adding up the length of sides of the kite. If a Kite ABCD is given then formula is given as

Perimeter of Kite ABCD = AB + BC + CD + AD

Read More,

• Real Life Examples of Kite
• Quadrilateral Formulas
• Types of Quadrilaterals
• Line of Symmetry

Summary:-

A kite is a type of quadrilateral with two distinct pairs of adjacent sides that are equal in length. The diagonals of a kite intersect at right angles, and one diagonal bisects the other. The area of a kite can be calculated using the formula Area= 1/2×Diagonal 1×Diagonal 2.

Solved Examples on Kite

Example 1: Find the Area of a Kite whose diagonals are 40 cm and 35 cm.

Solution:

Area of Kite with diagonals as d₁ and d₂ is given as 1/2 × d₁ × d₂

Given,

• d₁ = 40 cm
• d₂ = 35 cm

Area = 1/2 × 40 × 35

Area = 700 cm²

Hence, the area of kite is 700 cm²

Example 2: Find the unknown angles of the kite added in the image below,

Given,

• ∠JKL = 100°
• ∠KJL = 40°

👁 kite solved example 2

Solution:

We know that, main diagonal bisects the kite into two halves.

Thus,

∠KJL = ∠KLM

Hence ∠KLM = ∠KJL = 100° (given)

Sum of all angles of quadrilateral is 360°

Thus,

∠JML = 120°

Example 3: The Area of a kite-shaped field is 450 m² and the length of one of its diagonal is 50 m. A man wants to cross the field through the other diagonal. Find the distance the man has to travel.

Solution: 

Given,

• Area of Kite = 450 m²
• Length of One Diagonal(d₁) = 50 m

Let the other diagonal of kite is d₂

Area of Kite = 1/2 × d₁ × d₂

450 = 1/2 × 50 × d₂

d₂ = 18 m

Hence, other man has to travel a distance of 18 m.

Example 4: A park is shaped like a kite. If the diagonals of the park are 120 meters and 160 meters, how much fencing is needed to enclose the park if the sides measure 50 meters and 80 meters?

Solution:

The perimeter of the park (kite) can be calculated by summing the lengths of all sides: Perimeter=2×(50m+80m)=2×130m=260m

Therefore, 260 meters of fencing is needed.

Example 5:In a kite, one pair of opposite angles is 70° and 110°. Find the measure of the other pair of opposite angles.

Solution:

In a kite, the sum of all interior angles is 360°. Let the other two angles be x and x (since they are equal): 70°+110°+𝑥+𝑥=360°

Simplifying: 180°+2x=360°

Subtract 180° from both sides: 2x=180°

Divide by 2: x=90°

So, the other two angles are each 90°.

Example 6: Find the area of a kite with diagonals measuring 12 cm and 30 cm.

Solution:

To find the area of a kite, we use the formula:

Area = 1/2 × d1 × d2

Given:

• Diagonal d1 = 12 cm
• Diagonal d2 = 30 cm

Area = 1/2 × 12 × 30 = 180 cm²

Practice Questions on Kite

1. Find the area of kite whose one diagonal is 12 cm and other diagonal is 30 cm.
2. In a kite whose sides are, 5 cm and 8 cm. Find its perimeter.
3. Find the area of kite whose one diagonal is 7 cm and other diagonal is 3 cm.
4. In a kite with sides, 21 cm and 23 cm. Find its perimeter.
5. Define the term "kite" in geometry and list its main properties.
6. In a kite, if one diagonal is twice the length of the other, find the ratio of the areas of the two triangles formed by the diagonals.
7. Given a kite with sides of 5 cm, 5 cm, 8 cm, and 8 cm, calculate its perimeter.
8. Find the measure of the angles between the equal sides of a kite if the other two angles are 120° and 60°.
9. True or False: In a kite, the longer diagonal always bisects the shorter diagonal.
10. Find the area of a kite with diagonals measuring 15 cm and 20 cm.
11. Prove that in a kite, the diagonal that bisects the vertex angles is also the axis of symmetry.
12. How does the shape of a kite help in optimizing the design of a park with a central walking path along the diagonals?
13. If a kite has a perimeter of 48 cm and one pair of adjacent sides is 12 cm each, find the length of the other pair of adjacent sides.
14. A kite-shaped field has diagonals of 30 meters and 40 meters. Calculate the area of the field.

Conclusion

In summary, a kite is a unique quadrilateral with two pairs of equal adjacent sides and diagonals that intersect at right angles. Its distinct properties, such as symmetry and specific angle relationships, make it an important shape in geometry. The area and perimeter of a kite can be easily calculated using simple formulas, and understanding these properties is essential for solving various geometric problems and real-world applications.