Cyclic Quadrilateral - Definition, Properties, Examples

What is Cyclic Quadrilateral?

To understand the cyclic quadrilateral definition, let us break down the term. The word "cyclic" is related to a circle, and "quadrilateral" means a four-sided polygon. So, a quadrilateral of cyclic is a four-sided figure whose vertices all lie on the boundary of a single circle.

This circle is known as the circumcircle, and its points are referred to as concyclic points. A quadrilateral is cyclic if you can draw a circle that passes through each of its four corners. If even one point is inside or outside the circle, the figure does not satisfy the criteria.

Important Cyclic Quadrilateral Properties

The rules that come with this shape are what make it important for tests. These cyclic quadrilateral properties make it easier to find unknown angles and solve problems.

1. The Opposite Angles Rule

The most important rule is that the angles on opposite sides of a cyclic quadrilateral always add up to 180 degrees.

• Angle A + Angle C = 180 degrees

• Angle B + Angle D = 180 degrees

2. Exterior Angle Property

If one side of the quadrilateral is extended, an exterior angle is formed. In a quadrilateral of cyclic, this exterior angle is equal to the interior opposite angle. This property is very helpful in solving geometry proofs.

3. Area Calculation (Brahmagupta's Formula)

Whereas a general quadrilateral lacks an area formula, a cyclic quadrilateral does. If s is the semi-perimeter and the sides are a, b, c, and d, then:

Area = √(s−a)(s−b)(s−c)(s−d)

4. Ptolemy’s Theorem

The product of the diagonals in a cyclic quadrilateral is equal to the total of the products of the opposing sides. This is a unique thing about concyclic points: the way that sides and diagonals are related.

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Difference Between a Cyclic Quadrilateral and Non-Cyclic Quadrilateral

Here are some key differences to help identify them:

 Read More - Quadrilateral: Definition, Types, Properties, Examples

Practical Cyclic Quadrilateral Examples

To better understand what is cyclic quadrilateral, let us look at some situations.

Example 1: Square and Rectangle

All rectangles and squares are cyclic quadrilaterals. Since all angles are 90 degrees, opposite angles naturally add up to 180 degrees. A circle can always go through all four corners in this.

Example 2: The Isosceles Trapezium

An isosceles trapezium has two sides that are not parallel and are the same length. Because of this symmetry, all four corners can be on a circle. But a normal trapezium is not usually cyclic.

Example 3: Angle Calculation

Consider a quadrilateral of cyclic where Angle A is 70 degrees. Using the cyclic quadrilateral properties, the opposite angle (C) must be 110 degrees, since 180 − 70 = 110.

Solved Examples of Cyclic Quadrilateral

Problems involving a quadrilateral of cyclic often require careful reasoning. These cyclic quadrilateral examples will help you practise.

Example 1: Finding a Missing Opposite Angle

Problem: In a quadrilateral of cyclic ABCD, Angle A is 85°. Find Angle C.

Solution:
We know that opposite angles are supplementary.
Angle A + Angle C = 180°
85° + Angle C = 180°
Angle C = 95°

Example 2: Using the Exterior Angle Property

Problem: An exterior angle is 110°. Find the interior opposite angle.

Solution:
In a quadrilateral of cyclic, the exterior angle equals the interior opposite angle.
So, the required angle is 110°.

Example 3: Solving with Algebra

Problem: Opposite angles are (2x + 10)° and (x + 20)°. Find x and the angles.

Solution:
(2x + 10) + (x + 20) = 180
3x + 30 = 180
3x = 150
x = 50

Angles:
2(50) + 10 = 110°
50 + 20 = 70°

Example 4: Area Calculation

Problem: Sides are 3 cm, 4 cm, 5 cm, and 6 cm. Find the area.

Solution:
s = (3 + 4 + 5 + 6) / 2 = 9

Area = √(9−3)(9−4)(9−5)(9−6)
= √(6 × 5 × 4 × 3)
= √360 ≈ 18.97 cm²

Read More - Trapezium: Properties, Types, Formulas, and Applications

Importance of Cyclic Quadrilateral

Understanding quadrilateral of cyclic is useful beyond theory. It helps explain how shapes behave in different situations.

• Navigation: Many calculations use circle geometry.

• Architecture: These ideas are used in arches and round designs.

• Design: Concyclic points help make patterns and pictures that look good together.

How to Prove a Quadrilateral is Cyclic

To check if a quadrilateral is cyclic, use these methods:

• Verify if opposite angles add up to 180 degrees

• Check if the exterior angle equals the interior opposite angle

• See if perpendicular bisectors meet at one point (circle’s centre)