## Title: What Is Cyclic Quadrilateral

## Introduction:-

### Quadrilaterals are four-sided polygons that come in various shapes and sizes, each possessing unique properties and characteristics. Among them, the cyclic quadrilateral is a particularly intriguing geometric figure. A cyclic quadrilateral is a four-sided polygon whose vertices lie on the circumference of a single circle. In this article, I will try to explain What Is Cyclic Quadrilateral,its properties, and its different types, as well as explore their significance in geometry and real-world applications which is a very important concept for class (IX) and class (X) students.

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## Explanation:-

### To start this concept, let us first know

## What Is Cyclic Quadrilateral?

### To realise what a cyclic quadrilateral is, it's essential to break down the key components of the term:

### (1)Quadrilateral: A quadrilateral is a polygon with four sides. The sides are typically labeled as AB, BC, CD, and DA, while the vertices are labeled as A, B, C, and D.

### (2)Cyclic: The term "cyclic" refers to the property of the quadrilateral, which means that all four vertices of the quadrilateral lie on the circumference of a single circle.

### In the figure,'O' is a circle and ABCD is a cyclic quadrilateral.

In simple terms, a cyclic quadrilateral is a four-sided figure in which all its vertices touch the circumference of a circle. This geometric property sets cyclic quadrilaterals apart from other quadrilaterals.

## Properties of Cyclic Quadrilaterals:

### Cyclic quadrilaterals possess several interesting properties that make them worth studying. Let's explain some of the most important ones.

### 1.Sum of Opposite Angles: In a cyclic quadrilateral, the sum of the measures of two opposite angles is always equal to 180 degrees. This property is a consequence of the fact that the opposite sides of the quadrilateral are subtended by the same arc on the circumcircle.

### 2.Supplementary Adjacent Angles: The adjacent angles in a cyclic quadrilateral are supplementary, meaning they add up to 180 degrees. This property is true for any pair of angles that share a common side.

### 3.Diagonals' Intersection: The diagonals of a cyclic quadrilateral intersect at a point known as the "point of concurrence" or "cyclic quadrilateral theorem." This point lies on the circumcircle and has interesting implications in geometry.

### 4.Sum of Opposite Sides: The sum of the lengths of the opposite sides of a cyclic quadrilateral is constant. This property is related to Ptolemy's Theorem, which provides a relationship between the lengths of the sides and diagonals.

### 5.Exterior Angle Property: The exterior angle formed by extending one side of a cyclic quadrilateral is equal in measure to the opposite interior angle. This relationship can be helpful in solving problems involving cyclic quadrilaterals.

## Types of Cyclic Quadrilaterals:

### Cyclic quadrilaterals can be categorized into several types based on their properties and characteristics. Some of the most common types include:

### 1.Rectangle: A rectangle is a special type of cyclic quadrilateral in which all angles are right angles (90 degrees). Since a rectangle has four right angles, it can be inscribed in a circle.

### 2.Square: A square is a specific case of a rectangle where all sides are of equal length. Due to its right angles and equal sides, a square is also a cyclic quadrilateral.

### 3.Rhombus: A rhombus is a quadrilateral with all sides of equal length but not necessarily right angles. If the diagonals of a rhombus are equal in length, it becomes a cyclic quadrilateral.

### 4.Kite: A kite is a quadrilateral with two pairs of adjacent sides that are equal in length. If the diagonals of a kite are perpendicular, it is a cyclic quadrilateral.

### 5.Isosceles Trapezoid: An isosceles trapezoid is a quadrilateral with one pair of parallel sides and the other pair of non-parallel sides equal in length. It can be cyclic if the non-parallel sides are extended to meet.

## Some Theorems and problems based on cyclic quadralateral:-

### (1) Theorem: Prove that sum of opposite pair of angles of a cyclic quadrilateral is 180°

### Answer:- Let, ABCD is a cyclic quadrilateral of a circle with center 'O' To prove:- (i)<ABC + <ADC =180° and (ii) <BAD + <BCD = 180°

Construction:- OA and OC are joined.

Peoof:- <ADC and obtuse <AOC are the angles at the circumference and center respectively standing on the same arc ABC

∴ <ADC = ½ obtuse <AOC…..(1) [∵ standing on the same arc angle at the circumference is always half of the angle at the center]

Again ,

<ABC and reflex <AOC are the angles at the circumference and at the center standing at the same arc ADC

∴ <ABC = ½ reflex <AOC…..(2) [∵ standing on the same arc angle at the circumference is always half of the angle at the center]

Now (1)+(2) we get,

<ADC + <ABC = ½ ( obtuse <AOC + reflex <AOC)

= ½ complete <AOC

= (½ × 360)°

= 180°……(3)

Again,

ABCD is a quadrilateral,

∴<ADC + <ABC + <BAD + <BCD = 360° [ ∵ sum of all angles of a quadrilateral is 360°]

=> 180° + <BAD + <BCD = 360° [using equation(3)]

=> <BAD + <BCD = 360° - 180°

=> <BAD + <BCD = 180° Proved.

### (2) Find the radius of the circle, from the given picture whose <ABC = 150°and AC = 5 cm

### Solution:- This is a cyclic quadrilateral's picture.

### In the cyclic quadrilateral ABCD,

whose <ABC = 150° and AC = 5cm, To find, radius (r)=OA = OC =?

∵ In cyclic quadrilateral ABCD,

∵<ABC + <ADC = 180° [∵ sum of opposite pair of angles of a cyclic quadrilateral is 180°]

=> 150° + <ADC = 180°

=> <ADC = 180° - 150°

=> <ADC = 30°

Now, <ADC and <AOC are the angles at the circumference and at the center respectively.

∴ <AOC = 2 × <ADC [∵ standing on the same arc angle at the center is always double of the angle at the circumference]

= (2 × 30)°

= 60°

Again, OA = OC [ ∵ Radii of the same circle are equal]

=> <OCA = <OAC [ ∵ angles opposite to the equal sides are always equal]....(1)

### Now, In Δ AOC,

∵ <OCA + <OAC + <AOC = 180°

=><OCA + <OAC + 60° = 180°

=><OCA + <OAC = 180° - 60°

=><OCA + <OAC = 120°

=> <OCA + <OCA = 120° [using (1)]

=> 2 <OCA = 120°

=> <OCA = 120°/2

=> <OCA = 60°

∴ <OAC = 60° [ ∵ <OCA = <OAC]

Now In Δ AOC we find,

<OCA = <OAC = <AOC = 60°

∴ Δ AOC is an equilateral triangle

∴ OA = OC = AC

∴OA = OC =5cm [ ∵ AC = 5 cm given]

∴ Radius(r)of the circle = 5cm

### (3)Brahmagupta's Formula: Brahmagupta's Formula provides a method for calculating the area of a cyclic quadrilateral when the lengths of its sides and the length of one diagonal are known. It is a generalized version of Heron's Formula for triangles.

(4)Ptolemy's Theorem: Ptolemy's Theorem establishes a relationship between the sides and diagonals of a cyclic quadrilateral. It states that the product of the diagonals is equal to the sum of the products of the opposite sides.

(5)Inscribed Angle Theorem: The Inscribed Angle Theorem states that an angle formed by two chords in a circle is half the measure of the arc it subtends. This theorem is particularly useful when dealing with cyclic quadrilaterals because it helps determine angle measures.

## Applications of Cyclic Quadrilaterals:-

### While studying geometric shapes may seem abstract, cyclic quadrilaterals find practical applications in various fields, including mathematics, engineering, and even art.

Some real-world applications of cyclic quadrilaterals have been discussed below:-

### 1.Surveying and Navigation: Cyclic quadrilaterals are used in land surveying and navigation to calculate distances, angles, and positions accurately. Nautical charts and maps often employ these concepts to determine the location of objects or landmarks.

### 2.Engineering and Architecture: Architects and engineers use cyclic quadrilaterals to design structures and layouts, ensuring that angles and distances are accurate for stability and aesthetics. Roof trusses and building layouts often incorporate these principles.

### 3.Computer Graphics and Animation: In computer graphics and animation, cyclic quadrilaterals play a role in creating realistic and visually appealing 3D models and animations. Understanding the geometry of these shapes helps in rendering and animation design.

### 4.Astronomy: Cyclic quadrilaterals are essential in celestial navigation. Astronomers use them to calculate the positions of celestial bodies and their movement across the sky.

### 5.Art and Design: Artists and designers may incorporate geometric shapes, including cyclic quadrilaterals, into their work to achieve specific visual effects and compositions.

## Conclusion:-

### In this article I have tried to explain What Is Cyclic Quadrilateral,it's properties, some problems and theorems based on it.

Lastly we can say Cyclic quadrilaterals are fascinating geometric shapes that offer a wealth of mathematical properties, theorems, and practical applications. Their unique characteristic of having all four vertices on the circumference of a circle makes them a subject of great interest in the field of geometry. Understanding the properties and types of cyclic quadrilaterals, along with their real-world applications, can enhance one's mathematical knowledge and problem-solving skills. Whether any person a student, an engineer, an architect, or an artist, the study of cyclic quadrilaterals has something to offer, bridging the gap between theoretical mathematics and its...........

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FREQUENTLY ASKED QUESTIONs on Cyclic Quadrilateral:-
FAQ
(i).What is cyclic quadrilateral 9th?

+
Answer:A cyclic quadrilateral is a four-sided figure in which all its vertices touch the circumference of a circle. It is a two dimensional shape has four vertices and four edges.

(ii).Which shapes are cyclic quadrilateral?

+
Answer:A cyclic quadrilateral is in the following shapes:- (i) Square (ii)Rectangle (iii) Rhombus (iv)Isosceles trapezoid.

(iii).What is the definition of a cyclic quadrilateral?

+
Answer:In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a four-sided figure in which all its vertices touch the circumference of a circle. It is a two dimensional shape has four vertices and four edges.

(iv).What are the properties of quadrilaterals?

+
Answer:The properties quadrilaterals are:-
(i) The sum of all angles of a quadrilateral is 360°
(ii) A quadrilateral has two adjacent diagonals.
(iii) The diagonals of a quadrilateral bisect each-other.

(v).What if ABCD is a cyclic quadrilateral?

+
Answer:∵ ABCD is a cyclic quadrilateral, then CosA + CosB+ CosC + CosD = 0

Explanation:-

∵ ABCD is a cyclic quadrilateral,

then

A + C = 180° [∵ sum of opposite pair of angles of a cyclic quadrilateral is 180°]

=> A = 180° - C

=> CosA = Cos (180° - C)

=> CosA = - CosC

Again,

B + D = 180° [∵ sum of opposite pair of angles of a cyclic quadrilateral is 180°]

=> B = 180° - D

=> CosB = Cos (180° - D)

=> Cos B = - Cos D

Now,

CosA + CosB+ CosC + CosD

= - CosC - CosD + CosC + CosD [Putting values]

= 0

(i).What is cyclic quadrilateral 9th?

+Answer:A cyclic quadrilateral is a four-sided figure in which all its vertices touch the circumference of a circle. It is a two dimensional shape has four vertices and four edges.

(ii).Which shapes are cyclic quadrilateral?

+Answer:A cyclic quadrilateral is in the following shapes:- (i) Square (ii)Rectangle (iii) Rhombus (iv)Isosceles trapezoid.

(iii).What is the definition of a cyclic quadrilateral?

+Answer:In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a four-sided figure in which all its vertices touch the circumference of a circle. It is a two dimensional shape has four vertices and four edges.

(iv).What are the properties of quadrilaterals?

+Answer:The properties quadrilaterals are:- (i) The sum of all angles of a quadrilateral is 360° (ii) A quadrilateral has two adjacent diagonals. (iii) The diagonals of a quadrilateral bisect each-other.

(v).What if ABCD is a cyclic quadrilateral?

+Answer:∵ ABCD is a cyclic quadrilateral, then CosA + CosB+ CosC + CosD = 0

Explanation:-

∵ ABCD is a cyclic quadrilateral,

then

A + C = 180° [∵ sum of opposite pair of angles of a cyclic quadrilateral is 180°]

=> A = 180° - C

=> CosA = Cos (180° - C)

=> CosA = - CosC

Again,

B + D = 180° [∵ sum of opposite pair of angles of a cyclic quadrilateral is 180°]

=> B = 180° - D

=> CosB = Cos (180° - D)

=> Cos B = - Cos D

Now,

CosA + CosB+ CosC + CosD

= - CosC - CosD + CosC + CosD [Putting values]

= 0