[Maths Class Notes] on Cyclic Quadrilateral Pdf for Exam

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A four-sided polygon drawn in a circle is known as a cyclic quadrilateral. It has the largest area possible for a given side length. In other words, the quadrilateral inscribed in the circle depicts the largest area possible for these side lengths. In this article, we will know more about circular quadrilaterals, their theorems, and their properties. So, with no further ado, let us understand the cyclic quadrilateral in the coming section.

What is Cyclic Quadrilateral?

A cyclic quadrilateral is a quadrilateral that is surrounded by a circle. That is, a circle goes through each of the quadrilateral’s four vertices. A quadrilateral’s vertices are said to be concyclic. The centre of a circle is known as the circumcentre, and the radius is known as the circumradius. A circumcircle refers to a circle that contains all the vertices of any polygon on its circumference.

 

If all four vertices of the quadrilateral are on the circumference of the circle, it is called a cyclic quadrilateral. In other words, the vertices of a cyclic quadrilateral are formed by joining any four points on the circumference of a circle. It can be seen as a quadrilateral inscribed in a circle, with all four vertices of the quadrilateral lying on the circle’s circumference.  

                                                      

What is a Quadrilateral?

A polygon that has four sides, four corners, and four angles is known as a quadrilateral. The angles included are present at the four vertices of a quadrilateral. The word ‘quadrilateral’ is composed of two Latin words, Quadri, which means ‘four’, and latus, which means ‘side’. A quadrilateral is a two-dimensional figure having four edges. The sum of interior angles of a quadrilateral sums up to 360 degrees.

I.e. ∠A + ∠B + ∠C + ∠D = 360°

Properties of Cyclic Quadrilateral

Here’s a table that lists down the properties of a cyclic quadrilateral.

 

The opposite angles of a cyclic quadrilateral are supplementary which means that the sum of either pair of opposite angles is equal to 180 degrees.


The four perpendicular bisectors in a cyclic quadrilateral meet at the centre.


A quadrilateral is said to be cyclic if the sum of two opposite angles are supplementary.


The perimeter of a cyclic quadrilateral is 2s, where s = semi perimeter                       

s = [frac{a+b+c+d}{2}]


The area of a cyclic quadrilateral is  =[frac{1}{2}[s((s−a)(s−b)(s−c))]]

=[frac{1}{2}[s((s−a)(s−b)(s−c))]]

  where, a, b, c and d are the four sides of a quadrilateral.


A cyclic quadrilateral has four vertices which lie on the circumference of the circle.


If you just join the midpoints of the four sides in order in a cyclic quadrilateral, we get a rectangle or a parallelogram.


The perpendicular bisectors are concurrent in a cyclic quadrilateral.


If A, B, C and D are four sides of a quadrilateral and E is the point of intersection of the two diagonals in the cyclic quadrilateral then 

AE × EC = BE × ED

 

The Formula for the Perimeter of a Cyclic Quadrilateral 

The perimeter of a cyclic quadrilateral is 2s, where s = semi perimeter 

s = [frac{a+b+c+d}{2}]

Perimeter can be simplified in the following way,

Here, a+c = b+d,

          Substituting in the formula for semi perimeter we get, 

[s = frac{b+d+b+d}{2}]

[s=frac{2(b+d)}{2}]

s= b + d

The Formula for the Area of an Inscribed or Cyclic Quadrilateral 

The area of a cyclic quadrilateral is = [frac{1}{2}[s((s−a)(s−b)(s−c))]]

       =  [frac{1}{2}[s((s−a)(s−b)(s−c))]]

      where, a, b, c, and d are the four sides of a quadrilateral.

 

Cyclic Quadrilateral Theorem

Here is the important cyclic quadrilateral theorem.

  1. Theorem of Cyclic Quadrilateral (I) 

The either pair of the opposite angles of a cyclic quadrilateral sum up to 180°.

Given, ABCD is a cyclic quadrilateral of a cycle with the centre as O.

        Here, ∠BAD+∠BCD=180°

∠ABC+∠ADC=180°

Therefore, cyclic quadrilateral angles are equal to 180 degrees.

 

  1. Theorem of Cyclic Quadrilateral (II)

In a cyclic quadrilateral, if a quadrilateral is inscribed inside a cycle, the product of the diagonals of the cyclic quadrilateral is equal to the sum of the two pairs of opposite sides of the cyclic quadrilateral.                                                      

In a cyclic quadrilateral ABCD, AC and BD are diagonals and AB, CD, AD, and BC are opposite sides.

Product of Diagonals –

(AC×BD) = (AB×CD) +(AD×BC)

Ratio of Diagonals –

[frac{AC}{BD}=frac{(AB×AD) +(BC×CD)}{(AB×BC) +((AD×CD}]

 

Questions on Cyclic Quadrilateral Angles and Based on Cyclic Quadrilateral Theorem

Question 1: What will be the value of angle B of a cyclic quadrilateral if the value of angle D is equal to 60 degrees.

Solution: Let’s list down the given information, ∠B = 60°.

As quadrilateral ABCD is cyclic, which means that the sum of a pair of two opposite angles in a cyclic quadrilateral will be equal to 180° according to the cyclic quadrilateral theorem.

∠B+∠D = 180°

60°+∠D = 180°

∠D = 180° – 60°

Therefore, the value of ∠D = 120°.

 

Question 2: What will be the value of angle D of a cyclic quadrilateral if the value of angle B is equal to 70 degrees.

Solution:  Let’s list down the given information, ∠D = 70°.

As quadrilateral ABCD is cyclic, which means that the sum of a pair of two opposite angles in a cyclic quadrilateral will be equal to 180° according to the cyclic quadrilateral theorem.

∠B + ∠D = 180°

70° + ∠D = 180°

∠D = 180° – 70°

Therefore, the value of ∠D = 110°.

 

Question 3: Find the perimeter of a cyclic quadrilateral with sides 4 cm, 2 cm, 6 cm, and 8 cm. 

Solution:  Given the measurement of the sides are,

4 cm, 2 cm, 6 cm, and 8 cm.

Using the formula of the perimeter,

Perimeter = 2s

s = [frac{a+b+c+d}{2}]

s = [frac{4+6+2+8}{2}]

S = 10 

Therefore, the perimeter of a cyclic quadrilateral = 2s = 20.

We hope that this article about cyclic quadrilaterals has added value to your knowledge.

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