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 ↓


Arcs and Angles Subtended


Circles are one of the most fundamental shapes in geometry. They appear in nature, technology, and art, making their understanding important from both a theoretical and practical perspective. In this guide, we will discuss in depth one of the key concepts when dealing with circles: arcs and angles subtended.

What is a circle?

A circle is defined as the set of all points in a plane that are the same distance from a given point, called the center. This distance from the center to any point on the circle is called the radius.

Fundamentals of a circle

  • Radius - The distance from the center of a circle to any point on the circle.
  • Diameter - The longest distance from one end of a circle to the other. It is twice the length of the radius.
  • Circumference - The distance around the circle.
  • Arc - A portion of the circumference of a circle.
  • Sector - The area inside a circle bounded by two radii of the circle and the arc between them.
  • Central angle - The angle whose vertex is at the centre of the circle and whose sides are radii.

Understanding the arc

When you draw a curve between two points on a circle, this curve is called an arc. Arcs are represented by length in terms of their endpoints and the angle subtended at the center.

Arch

The red curve in the picture above is the arc of the circle. You have two types of arcs:

  • Minor arc - An arc that is smaller than a semicircle.
  • Major arc - An arc that is larger than a semicircle.

Angle subtended by the arc at the centre

The angle subtended by an arc at the centre of a circle is called the central angle. It is formed by drawing two radii from the centre to the end points of the arc.

Arc AB subtends a central angle ∠AOB.
Where O is the centre of the circle.
∠AOB

In this diagram, the arc from point A to point B forms the central angle ∠AOB. Remember, central angles are always measured in degrees.

Angle subtended by an arc on a circle

The angles formed by the same arc at different points on a circle are called angles in the same line segment. These angles are equal to each other.

∠ACB ∠ADB

Note that both ∠ACB and ∠ADB are subtended by the same arc AB and are equal.

Important theorems

Angle theorem at the center

The central angle is always twice any angle subtended by the same arc on the circumference. Thus, if θ is the angle subtended on the circle, then the central angle is 2θ.

∠AOB = 2 * ∠ACB

Example problems

Example 1:

Consider a circle with centre O. Let points A and B lie on the circle such that the arc AB subtends the central angle ∠AOB = 80° at centre O. Find the angle subtended by the same arc at point C on the circle.

The angle subtended by the arc at any point on the circle is half the angle subtended at the centre.

∠ACB = ∠AOB / 2 = 80° / 2 = 40°

Example 2:

In a given circle, arc XY subtends an angle of 100° at the centre O. Find the angle subtended by arc XY at any point Z on the opposite segment of the circle.

∠XZY = ∠XOY / 2 = 100° / 2 = 50°

Practice problem 1:

If the angle subtended by an arc at the centre of a circle is 120°, then what will be the angle subtended by the same arc at any point on the circle?

Solution:
∠A = 120° / 2 = 60°

Practice problem 2:

A chord divides a circle into two arcs, major and minor. If an arc subtends an angle of 130° at the centre of the circle, find both the angles subtended by the arc at any point on the circle which does not lie on the diameter.

Solution:
Angle subtended at any point for minor arc = 130° / 2 = 65°
Angle subtended at any point for major arc = 360° - 130° = 230°
230° / 2 = 115°

Conclusion

Understanding the relationship between arcs and the angles they form is an important part of geometrical studies. When examining circles, the link between central angles and the angles formed at the circumference serves as a bridge to connect various geometrical theorems and properties. Practicing these concepts through the given exercises and visualizations helps strengthen your understanding of this essential topic in mathematics.


Grade 9 → 10.6


U
username
0%
completed in Grade 9

← Prev (10.5)
Cyclic Quadrilaterals
Next (11) →
Constructions

Comments 



Arcs and Angles Subtended

https://www.buddymath.com/g9/10.6?bmal=en