Grade 10
  1. Number Systems  1.1. Real Numbers  1.1.1. Properties of Real Numbers  1.1.2. Rational and Irrational Numbers  1.1.3. Decimal Representation of Real Numbers  1.1.4. Operations on Real Numbers  1.1.5. Real Numbers on the Number Line  1.2. Euclid's Division Lemma  1.3. Prime Factorization  1.4. LCM and HCF  1.5. Exponents and Radicals  1.5.1. Laws of Exponents  1.5.2. Simplifying Radical Expressions  1.5.3. Scientific Notation
  2. Algebra  2.1. Polynomials  2.1.1. Definition and Types of Polynomials  2.1.2. Degree of a Polynomial  2.1.3. Zeros of a Polynomial  2.1.4. Factorization of Polynomials  2.1.5. Algebraic Identities  2.2. Linear Equations in Two Variables  2.2.1. Graph of Linear Equations  2.2.2. Solutions of Linear Equations  2.2.3. Application of Linear Equations in Word Problems  2.3. Quadratic Equations  2.3.1. Standard Form of a Quadratic Equation  2.3.2. Methods of Solving Quadratic Equations  2.3.2.1. Factorization Method  2.3.2.2. Completing the Square Method  2.3.2.3. Quadratic Formula  2.3.3. Nature of Roots  2.4. Arithmetic Progressions  2.4.1. Definition of Arithmetic Progression  2.4.2. General Term of an AP  2.4.3. Sum of First n Terms of an AP  2.5. Introduction to Functions  2.5.1. Definition and Types of Functions  2.5.2. Domain and Range  2.5.3. Composition of Functions  2.5.4. Inverse of a Function
  3. Coordinate Geometry  3.1. Cartesian System  3.2. Distance Formula  3.3. Section Formula  3.4. Area of a Triangle  3.5. Slope of a Line  3.6. Equation of a Line  3.6.1. Slope-Intercept Form  3.6.2. Point-Slope Form  3.6.3. Two-Point Form  3.6.4. Intercept Form  3.6.5. General Form of a Line
  4. Trigonometry  4.1. Trigonometric Ratios  4.1.1. Sine Cosine Tangent and their Reciprocals  4.1.2. Values of Trigonometric Ratios at Specific Angles  4.2. Trigonometric Identities  4.3. Trigonometric Ratios of Complementary Angles  4.4. Heights and Distances  4.4.1. Angle of Elevation and Depression  4.4.2. Applications in Real-Life Problems
  5. Geometry  5.1. Triangles  5.1.1. Congruence of Triangles  5.1.2. Criteria for Congruence  5.1.3. Properties of Triangles  5.2. Similarity  5.2.1. Similar Figures  5.2.2. Criteria for Similarity  5.2.3. Similarity of Triangles  5.2.4. Applications of Similarity in Real-Life  5.3. Circles  5.3.1. Properties of a Circle  5.3.2. Tangent to a Circle  5.3.3. Number of Tangents from a Point  5.3.4. Angle Subtended by an Arc  5.4. Constructions  5.4.1. Construction of Similar Triangles  5.4.2. Tangent to a Circle  5.4.3. Construction of Angle Bisectors
  6. Mensuration  6.1. Areas of Plane Figures  6.1.1. Area of a Triangle  6.1.2. Area of a Parallelogram  6.1.3. Area of a Trapezium  6.1.4. Area of a Rhombus  6.2. Surface Areas and Volumes  6.2.1. Surface Area of a Cube Cuboid and Cylinder  6.2.2. Surface Area of a Cone and Sphere  6.2.3. Volume of a Cube Cuboid and Cylinder  6.2.4. Volume of a Cone and Sphere  6.3. Conversion of Units  6.4. Frustum of a Cone
  7. Statistics  7.1. Introduction to Statistics  7.2. Collection of Data  7.3. Presentation of Data  7.3.1. Tabular Form  7.3.2. Graphical Form  7.3.2.1. Bar Graph  7.3.2.2. Histogram  7.3.2.3. Frequency Polygon  7.3.2.4. Ogive  7.4. Measures of Central Tendency  7.4.1. Mean Median and Mode  7.4.2. Quartiles and Percentiles  7.5. Measures of Dispersion  7.5.1. Range and Interquartile Range  7.5.2. Standard Deviation and Variance
  8. Probability  8.1. Introduction to Probability  8.2. Experimental Probability  8.3. Theoretical Probability  8.4. Probability of Simple Events  8.5. Complementary Events  8.6. Probability of Compound Events

Grade 10 → Geometry → Circles ↓


Properties of a Circle


A circle is a fundamental geometric figure that is defined as the set of all points in a plane that are equidistant from a given point, called the center. The properties of a circle are essential to understand its behavior, measurement, and relationships. Let us explore the various properties of a circle in detail.

Basic components of a circle

Before diving into the properties, it is important to understand the basic components of a circle:

  • Centre: The centre is the fixed point around which the circle is drawn. It is usually denoted by C
  • Radius: The radius of a circle is any line segment from its center to the circumference. It is usually denoted by r.
  • Diameter: The diameter is a line segment that passes through the center and has an endpoint on the circle. It is the longest distance across the circle and is twice the radius: d = 2r.
  • Perimeter: The circumference is the perimeter or boundary of a circle. It is calculated using the formula: C = 2πr.
  • Area: The space contained within a circle is its area, given by the formula: A = πr².

Essential properties of a circle

Now that we understand the basic components, let's look at the essential properties of a circle:

Perfect symmetry of a circle

The circle is a perfectly symmetrical figure. This symmetry means that every point on the edge of the circle is the same distance from the center. This property allows the circle to have infinite symmetry lines, since any diameter divides the circle into two equal halves.

Circumscribed and inscribed circles

A circle can be both circumscribed and inscribed. A circumscribed circle is one that passes through all the vertices of a polygon. In contrast, an inscribed circle is one that lies within a polygon and touches every side.

Angles in a circle

The central angle is the angle whose vertex is the center of the circle. It can be associated with the length of the arc - part of the circumference of the circle. The whole circle forms a total angle of 360 degrees.

Example of an angle

Consider a circle with center at C If a point A and another point B on the circle subtend an angle at C, then this angle is the central angle. If the arc from A to B goes around the circle and covers 180 degrees, then the central angle at C is also 180 degrees.

Visual example

The basic structure of the circle

R D C

The figure above shows a circle with center C The red line is the diameter d, and the blue line is the radius r.

Perimeter and area

C(circumference) a = πr²

In the above diagram, the green line represents the circumference C of the circle. The area inside the circle is given by A = πr².

Special theorems related to circles

Inscribed angle theorem

The inscribed angle theorem states that the inscribed angle in a circle is half the measure of the central angle that forms - or holds - the same arc. This means that if you have an inscribed angle and you know which arc it is "holding", then you know that the central angle that forms this arc is twice the inscribed angle.

Example of inscribed angle theorem

Suppose you have a circle with an angle of ∠ABC and an arc AC. If the central angle ∠AOC forms the same arc, then the measure of ∠AOC is twice the measure of ∠ABC.

Tangent and secant properties

A tangent is a line that touches a circle at exactly one point. A secant is a line that intersects a circle at two points. Properties associated with tangents and secants include:

  • Tangent-secant theorem: If a tangent and a secant line are drawn from a point outside a circle, then the square of the length of the tangent segment is equal to the product of the length of the outer part of the secant segment and the whole length.
  • Angle between tangents: The angle between two tangents drawn from an external point is half the difference in the measure of the intercepted arcs.

More advanced examples

To further understand the properties of circles, consider the following examples that illustrate these concepts.

Example 1: Calculating circle measurement

A circle with radius r = 5 units is given:
- Diameter: d = 2r = 10 units
- Circumference: C = 2πr ≈ 31.42 units
- Area: A = πr² ≈ 78.54 square units

Example 2: Angle measurement

A circle has an arc of 120 degrees. An inscribed angle ∠ABC forming this arc measures half of the arc angle, i.e. 60 degrees.

Example 3: Tangent-jump theorem

From a point P outside a circle, a tangent PT measures 8 units, and a secant PQ cuts the circle such that PQ = 12 and QR = 4 According to the tangent-edge theorem:
PT² = PQ * PR
8² = 12 * (12 + 4)
64 = 12 * 16

Conclusion

The properties of a circle provide interesting information about this fundamental geometric shape. Recognizing these properties not only enhances problem-solving skills but also deepens the understanding of geometry. Circles form the basis of many concepts and applications in mathematics, engineering, and science.


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Properties of a Circle

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