Grade 9
  1. Number Systems  1.1. Real Numbers  1.2. Rational Numbers  1.3. Irrational Numbers  1.4. Properties of Real Numbers  1.5. Laws of Exponents and Surds  1.6. Representation on the Number Line  1.7. Decimal Representation of Real Numbers
  2. Polynomials  2.1. Definition and Types of Polynomials  2.2. Degree of a Polynomial  2.3. Zeroes of a Polynomial  2.4. Remainder Theorem  2.5. Factorization of Polynomials  2.6. Algebraic Identities  2.7. Polynomial Equations and Roots
  3. Coordinate Geometry  3.1. Cartesian System  3.2. Plotting Points on the Plane  3.3. Distance Formula  3.4. Section Formula  3.5. Area of a Triangle  3.6. Coordinates of Midpoint and Centroid
  4. Linear Equations in Two Variables  4.1. Solutions of a Linear Equation  4.2. Graphical Method  4.3. Algebraic Method  4.4. Applications of Linear Equations  4.5. Linear Inequalities
  5. Introduction to Euclidean Geometry  5.1. Basic Definitions and Terms  5.2. Axioms and Postulates  5.3. Theorems on Angles and Lines  5.4. Properties of Parallel Lines  5.5. Construction of Triangles  5.6. Congruence Axioms
  6. Lines and Angles  6.1. Angle Pair Relationships  6.2. Parallel Lines and Transversal  6.3. Properties of Angles Formed by Parallel Lines  6.4. Angle Sum Property of a Triangle  6.5. Types of Angles
  7. Triangles  7.1. Congruence of Triangles  7.2. Criteria for Congruence  7.3. Properties of Triangles  7.4. Inequalities in a Triangle  7.5. Similarity of Triangles  7.6. Pythagorean Theorem
  8. Quadrilaterals  8.1. Properties of Quadrilaterals  8.2. Types of Quadrilaterals  8.3. Midpoint Theorem  8.4. Properties of Parallelograms  8.5. Trapezium and Kite Properties  8.6. Diagonals and their Properties
  9. Areas of Parallelograms and Triangles  9.1. Area of a Parallelogram  9.2. Area of a Triangle  9.3. Proofs of Area Theorems  9.4. Applications of Area Theorems
  10. Circles  10.1. Definitions and Terms  10.2. Properties of a Circle  10.3. Chord Properties  10.4. Tangents to a Circle  10.5. Cyclic Quadrilaterals  10.6. Arcs and Angles Subtended
  11. Constructions  11.1. Basic Constructions  11.2. Constructing Bisectors  11.3. Constructing Triangles  11.4. Constructing Tangents to a Circle  11.5. Constructing Angles of Given Measure
  12. Heron's Formula  12.1. Area of a Triangle Using Heron’s Formula  12.2. Application to Different Triangles and Quadrilaterals  12.3. Proof of Heron’s Formula
  13. Surface Areas and Volumes  13.1. Surface Area of a Cube Cuboid and Cylinder  13.2. Volume of a Cube Cuboid and Cylinder  13.3. Surface Area and Volume of a Cone  13.4. Surface Area and Volume of a Sphere and Hemisphere  13.5. Conversion of Units  13.6. Volume of a Prism
  14. Statistics  14.1. Collection of Data  14.2. Presentation of Data  14.3. Measures of Central Tendency  14.4. Mean Median and Mode  14.5. Graphical Representation of Data  14.6. Range and Measures of Dispersion
  15. Probability  15.1. Introduction to Probability  15.2. Experimental Probability  15.3. Theoretical Probability  15.4. Simple Problems on Probability

Grade 9 → Circles ↓


Cyclic Quadrilaterals


A cyclic quadrilateral is a special type of quadrilateral whose all vertices lie on the circumference of a circle. This circle is called the circumcircle of the quadrilateral, and the quadrilateral is inscribed in the circle. Cyclic quadrilaterals are an important concept in geometry in understanding the relationship between circles and quadrilaterals.

Understanding cyclic quadrilaterals

Every cyclic quadrilateral has certain unique properties. Before considering these properties in detail, let us consider the basic structure of a cyclic quadrilateral.

  A quadrilateral is a four-sided polygon with four vertices and four angles. Let us consider such a quadrilateral ABCD, where points A, B, C and D are points lying on a circle.

The concept of a cyclic quadrilateral allows us to explore properties related to the angles and sides of a quadrilateral, as well as its circumcircle.

Properties of cyclic quadrilaterals

Some notable properties make cyclic quadrilaterals distinct from other quadrilaterals:

  1. Opposite angles are supplementary: One of the basic properties of cyclic quadrilaterals is that opposite angles are supplementary. This means: 
        ∠A + ∠C = 180°
    ∠B + ∠D = 180°
    This property can be easily derived using the inverted angle theorem of a circle.
  2. Ptolemy's theorem: For a cyclic quadrilateral, Ptolemy's theorem states that the product of its diagonals is equal to the sum of the products of the pairs of its two opposite sides: 
        AC * BD = AB * CD + AD * BC
  3. Angle Bisector Property: In a cyclic quadrilateral, the angles formed by angle bisectors of opposite angles are equal.

Visual representation

Let's draw a cyclic quadrilateral in the coordinate system to better understand its structure and properties:

A B C D

In the above figure, cyclic quadrilateral ABCD is inscribed in a circle with center O. The vertices A, B, C and D lie on the circumference of the circle.

Examples and exercises

Let us consider some examples and exercises to understand the applications of cyclic quadrilateral properties:

Example 1:

Consider a cyclic quadrilateral ABCD where ∠A = 70° and ∠B = 80°. Find the measures of ∠C and ∠D.

Using the property that opposite angles are supplementary in a cyclic quadrilateral:

∠A + ∠C = 180°
∠C = 180° - ∠A
∠C = 180° - 70° = 110°

Similarly,
∠B + ∠D = 180°
∠D = 180° - ∠B
∠D = 180° - 80° = 100°

Hence, ∠C = 110° and ∠D = 100°.

Example 2:

In a cyclic quadrilateral EFGH, if EF = 3 cm, FG = 4 cm, GH = 5 cm, and HE = 6 cm, and one diagonal EG = 7 cm, then find the other diagonal FH using Ptolemy's theorem.

According to Ptolemy's theorem for a cyclic quadrilateral:

EG * FH = EF * GH + FG * HE

Insert known values:
7 * FH = 3*5 + 4*6

7 * FH = 15 + 24
7 * FH = 39
FH = 39 / 7 = 5.57 cm

Applications of cyclic quadrilaterals

The study of cyclic quadrilaterals has various applications in geometry problems and proofs. They help solve various problems involving angles and distances around circles. The properties of cyclic quadrilaterals are also used in the fields of trigonometry and can be applied in calculating unknowns in geometry-based problems.

In addition, cyclic quadrilaterals appear in the study of advanced geometry, such as the design of cyclic polygons and investigations into exocircles and incenters of triangles. Their properties provide useful tools for solving complex problems that involve circular shapes.

Conclusion

The study of cyclic quadrilaterals enhances the fundamental understanding of geometric properties and relationships in circular geometry. By understanding that all the vertices of such a four-sided figure lie on a circle, students can use the unique properties of cyclic quadrilaterals to solve a variety of mathematical problems. A strong understanding of cyclic quadrilaterals enriches students' understanding of both basic and advanced geometry.


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Cyclic Quadrilaterals

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