Exploring Kite and its Theorems – Properties, examples, and Application

The Kite and its Theorems are geometric properties that apply specifically to the kite shape.

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.

Let's explore The Kite and its Theorems in detail.....

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.

Let’s look into more details of each theorem with examples:

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.

Theorem 1 : If a quadrilateral is a kite, then its diagonals are perpendicular.

GIVEN: AB ≅ CB and AD ≅ CD
PROVE THAT: AC ⊥ BD
Proof :

Opposite angles are congruent : If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent.

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.

Example based on kite and its theorems :

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.

Learning about kites and their theorems and properties can be valuable for several reasons:

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.

The properties and theorems of kites find applications in various real-world scenarios. Here are a few examples:

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.

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