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• If the four vertices of a quadrilateral lie on the circumference of the circle then it is known as a cyclic quadrilateral.

• A cyclic quadrilateral is also known as a circumcircle or a circumscribed cycle.

• The vertices of a quadrilateral are said to be concyclic.

• A quadrilateral is a polygon having four edges and four corners or vertices.

• The sum of interior angles of a quadrilateral sum up to 360 degrees.

• 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.

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 = $= \frac{1}{2}\left[ {s\left( {\left( {s - a} \right)\left( {s - b} \right)\left( {s - c} \right)} \right)} \right]$      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

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\left( {b + d} \right)}}{2}$s= b + d

Formula for the Area of an Inscribed or Cyclic Quadrilateral -

 The area of a cyclic quadrilateral is = $= \frac{1}{2}\left[ {s\left( {\left( {s - a} \right)\left( {s - b} \right)\left( {s - c} \right)} \right)} \right]$      where, a, b, c and d are the four sides of a quadrilateral.

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, $\angle BAD + \angle BCD = {180^o}$

$\angle ABC + \angle ADC = {180^o}$

Therefore, cyclic quadrilateral angles 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 -

$\left( {AC \times BD} \right){\text{ }} = {\text{ }}\left( {AB \times CD} \right){\text{ }} + \left( {AD \times BC} \right)$

Ratio of Diagonals -

$\frac{{AC}}{{BD}} = \frac{{\left( {AB \times AD} \right) + \left( {BC \times CD} \right)}}{{\left( {AB \times BC} \right) + \left( {AD \times CD} \right)}}$

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, $\angle B$ = 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.

$\angle B$+$\angle D$ = 1800

600+ $\angle D$ = 1800

$\angle D$ = 1800 - 600

Therefore, the value of$\angle D$ = 1200

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, $\angle D = {70^o}$

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

$\angle B$ + $\angle D$ = 1800

$\angle B$ + 700 = 1800

$\angle B$ = 1800 - 700

Therefore, the value of$\angle D = {110^o}$

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

Solution:  Given the measurement of the sides are,

4cm, 2 cm, 6 cm and 8 cm

Using the formula of perimeter,

Perimeter = 2s

$S = \frac{{a + b + c + d}}{2}$

$S = \frac{{4 + 6 + 2 + 8}}{2}$

s= 10

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

Question 1) What are the Properties of Cyclic Quadrilaterals?

## The Properties of a Cyclic Quadrilateral are Listed Below -

Question 2) What is a Cyclic Quadrilateral?

Answer) A four-sided geometric figure whose vertices lie on the circumference of the circle is known as a cyclic quadrilateral.

Question 3) Are all Rectangles Cyclic?

Answer) Yes, all rectangles are cyclic, but many quadrilaterals are not cyclic. As we know that if the opposite angles of a quadrilateral are supplementary which means that the sum of either pair of opposite angles is equal to 180 degrees only then it is cyclic.