Kites are a type of quadrilateral that have two pairs of adjacent sides that are congruent. In addition, their diagonals intersect at right angles and longer diagonal bisect the shorter diagonal. One pair of opposite angles are congruent. These properties give rise to several important theorems that have significant real-world applications.

**Diagonal
Properties:** A kite has two diagonals, one that connects opposite corners and
another that connects the other two opposite corners. The first theorem of kite
states that the diagonals of a kite are perpendicular, meaning they intersect
at a 90-degree angle.

AC ⊥BD

∡AOB=∡AOD=∡BOC=∡DOC = 90°

Moreover, the second theorem states that the longer diagonal of a kite bisect the shorter diagonal

Longer diagonal AC bisects the shorter diagonal BD.

BO ≌OD

BO = OD= ½ BD

**Angle
Bisector Theorem:** If one of the diagonals of a kite bisects an angle of the
kite, then it also bisects the opposite angle of the kite.

**Equal Length
Theorem:** A kite has two pairs of adjacent sides that are equal in length.

AB ≌AD

BC ≌ DC

The third theorem of kite states that the longer diagonal of a kite bisects the angle between the two pairs of adjacent sides that are equal in length.

AC bisects ∡BAD =>∡BAC =∡CAD

AC bisects ∡BCD => ∡BCA = ∡DCA

**One pair of opposite angles are
congruent:** The fourth theorem of kite is that one pair of opposite angles are
congruent.

∡B ≌∡D

**Midpoint
Theorem:** The fifth theorem of kite states that the line segment connecting the
midpoints of the non-parallel sides of a kite is perpendicular to the line
segment connecting the endpoints of the shorter pair of adjacent sides.

P,Q,R and S are the mid points of the sides of the kite.

So ∡P= ∡Q =∡R=∡S = 90^0

*These
theorems are essential in solving problems that involve kites and their
properties. They can be used to find missing side lengths or angles, determine
the area of a kite, or prove geometric relationships between different parts of
the kite.*

**Diagonal
Properties:** A kite has two diagonals, one that connects opposite corners and
another that connects the other two opposite corners. The first theorem of kite
states that the diagonals of a kite are perpendicular, meaning they intersect
at a 90-degree angle. Moreover, the second theorem states that the diagonals of
a kite bisect each other, meaning they divide each other into two equal segments.

GIVEN: AB ≅ CB and AD ≅ CD

PROVE THAT: AC ⊥ BD**Proof :**

Given: ABCD is a kite with AB ≅ AD and BC ≅ CD.

Prove that:∠A ≅ ∠C

**Proof :**

Given: ABCD is a kite with AB ≅ AD and BC ≅ CD.

Prove that:∠A ≅ ∠C.

**Proof :**

*In
summary, the Angle Bisector Theorem for kites states that if one diagonal of a
kite bisects an angle of the kite, then it also bisects the opposite angle of
the kite. This property is useful in solving geometric problems involving
kites, where the bisecting diagonal can be used to find missing angles or side
lengths.*

The Equal Length Theorem for kites states that if a kite has two pairs of adjacent congruent sides, then its diagonals are congruent.

The Equal Length Theorem for kites states that if a kite has two pairs of adjacent congruent sides, then its diagonals are congruent. This theorem can be useful in identifying congruent segments in a kite, which can help in solving geometric problems involving kites.

The Kite Midpoint Theorem states that in a kite, the line segment connecting the midpoints of the two non-adjacent sides is perpendicular to the line segment connecting the endpoints of the other diagonal, and it bisects it. This theorem can be useful in solving geometric problems involving kites, where the midpoint of a diagonal needs to be found or a perpendicular bisector needs to be drawn.

1) In a kite, ABCD,AB = x + 2 , BC = 2x + 1. The perimeter of the kite is 48cm. Find x and also find the length of each side.

**Solution :**

As we know that, adjacent sides in a kite are
equal.

∴ AB = AD and BC = CD.

Perimeter = sum of all the sides

P = AB + BC + CD + AD

48 = x + 2 + 2x + 1 + x + 2 + 2x + 1

48 = 6x + 6

⇒ 6x = 48 -6

∴ 6x = 42

x = 42/6

x = 7

∴ AB = AD = x + 2 = 7 + 2 = 9cm

and BC = CD = 2x + 1 = 2(7) + 1 = 14 + 1 = 15 cm

2) Find the area of kite when the diagonals are 12cm and 18cm.

**Solution:**
Area of kite = ½ (product of two diagonals)

Area = ½ ( d1 x d2)

= ½ (12 x 18)

= ½ (216)

Area = 108 cm²

3) In kite WEAR, ∠WEA = 70° and ∠ARW = 80°. Find the remaining two angles.

Solution : ∠WEA = 70° so ∡WER = ∡AER = 35°

∠ARW = 80°, so ∡ERW = ∡ERA = 40°

In ∆WRE, ∡W + ∡WER + ∡ERW=180

∡W + 35 + 40 =180

∡W +75 =180

∡W= 180 -75

∡W= 105°

∡W =∡A

∡A = 105°

One of the key uses of the Kite Theorem is in geometry, where it can be used to prove various other theorems and solve problems involving quadrilaterals. For example, the Kite Midpoint Theorem, which states that the line segment connecting the midpoints of the two non-adjacent sides of a kite is perpendicular to the diagonal that connects the endpoints of the other diagonal and bisects it, can be used to find the length of a diagonal or to draw a perpendicular bisector.

**1.Geometric understanding: **Studying kites helps
develop a deeper understanding of geometric concepts. Kites are a specific type
of quadrilateral with unique properties, and exploring these properties
enhances geometric reasoning skills. It allows you to grasp the relationships
between angles, sides, and diagonals in a kite, contributing to a broader
understanding of shape and form.

**2.Problem-solving abilities:** Learning about kites
and their theorems equips you with problem-solving skills. By working with the
properties and theorems of kites, you develop logical reasoning and analytical
thinking. These skills are transferable to various mathematical and real-world
scenarios, where problem-solving skills are essential.

**3.Practical applications: **Understanding the
properties of kites can have real-world applications, as mentioned earlier.
From architecture and engineering to kite sports and recreational activities,
the knowledge of kite properties can be useful in designing structures,
understanding aerodynamics, or participating in kite-related hobbies.

**4.Mathematical connections:** Studying kites
connects with other areas of mathematics. Kite theorems and properties are
often linked to concepts such as congruence, similarity, trigonometry, and
coordinate geometry. By exploring kites, you strengthen your overall
mathematical foundation and see how different concepts interrelate.

**5.Intellectual curiosity:** Learning about kites can
spark curiosity and fascination. Geometric shapes have aesthetic appeal and can
captivate the mind. Discovering the intricacies of kites and their properties
can be intellectually stimulating and foster a passion for mathematics and
geometry.

**6.Educational foundation:** Kites are part of the
mathematical curriculum in many educational systems. Learning about kites and
their theorems provides a solid foundation for further mathematical studies. It
sets the stage for exploring more complex concepts and geometrical shapes in
higher-level mathematics.

*In
summary, learning about kites and their theorems and properties enriches your
understanding of geometry, enhances problem-solving skills, offers practical
applications, strengthens mathematical connections, satisfies intellectual
curiosity, and provides a foundation for future mathematical studies.*

**1.Construction
and design: **Kites are used in the design and construction of various objects.
For example, in architecture and engineering, the properties of kites can be
applied when designing trusses or frameworks for buildings and bridges. Kites
provide stability and distribute forces evenly across their structure.

**2.Aerospace
engineering:** The shape and properties of kites are utilized in the design of
certain aircraft, such as kite planes or kite-powered systems. Kites can
generate lift, and their stability can be utilized to enhance maneuverability and control.

**3.Kiteboarding and kite surfing:** These water sports activities involve the use of
kites to harness wind power and propel riders across the water. The aerodynamic
properties of kites are crucial in achieving lift and controlling the direction
and speed of movement.

**4.Traction
kites:** Large kites known as traction kites are used for activities like kite
buggying, kite landboarding, and kite skiing. These kites generate significant
power and are harnessed by riders to pull them along the ground or snow.

**5.Recreational
kites: **Traditional diamond-shaped kites or other variations are commonly used
for recreational purposes. People fly kites in parks, beaches, and open spaces
for enjoyment and relaxation.

*These are
just a few examples of how the properties and theorems of kites find
applications in the real world. Kites have been used for various practical
purposes throughout history, and their versatility continues to make them
useful in different fields.*

• Introduction to Quadrilateral

• Types of Quadrilateral

• Properties of Quadrilateral

• Parallelogram and its Theorems

• Rectangle and its Theorems

• Square and its Theorems

• Rhombus and its Theorems

• Trapezoid (Trapezium)and its Theorems

• Kite and its Theorems

• Mid Point Theorem

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