What is Circles Class 9: Definition & Key Concepts Explained

Definition and Basic Elements of a Circle

A circle is defined as the set of all points in a plane that are at a fixed distance from a fixed point called the centre.

Key elements of a circle:

• Centre (O): The fixed point from which all points on the circle are equidistant.
• Radius (r): The distance from the centre to any point on the circle.
• Diameter (d): A chord passing through the centre; the longest chord in the circle.
• Chord: A line segment with both endpoints on the circle.
• Arc: A part of the circumference of the circle.
• Circumference: The complete boundary line of the circle.
• Tangent: A line that touches the circle at exactly one point.

Understanding these basic terms is crucial for solving problems related to circles in Class 9 NCERT Maths.

Properties of Circles You Must Know

Several important properties govern the behaviour and relationships of parts of a circle:

• The radius is perpendicular to the tangent at the point of contact.
• All radii of a circle are equal.
• The diameter is twice the radius, mathematically expressed as:

$$ d = 2r $$

• A chord that passes through the centre is the longest chord.
• Equal chords are equidistant from the centre.
• The perpendicular bisector of a chord passes through the centre of the circle.

These properties help in proving theorems and solving geometry problems in Class 9.

Difference Between Radius, Diameter, and Chord

Understanding the difference between radius, diameter, and chord is essential:

Example: If the radius of a circle is 7 cm, then the diameter is:

$$ d = 2 \times 7 = 14 \text{ cm} $$

This distinction is frequently tested in Class 9 exams.

Understanding Tangents and Their Properties

A tangent to a circle is a straight line that touches the circle at exactly one point, called the point of contact.

Key properties of tangents:

• The tangent is perpendicular to the radius drawn to the point of contact.
• A circle can have infinitely many tangents, but only one tangent at a particular point.
• Two tangents drawn from an external point to a circle are equal in length.

Formula: If $PT$ is a tangent and $OP$ is the radius, then:

$$ OP \perp PT $$

where $O$ is the centre and $T$ is the point of contact.

Tangents are important in solving geometry problems involving circles in Class 9.

Worked Example: Finding the Diameter from Radius

Problem: Find the diameter of a circle if its radius is 10 cm.

Solution:

Given radius, $r = 10$ cm

We know diameter $d = 2r$

So,

$$ d = 2 \times 10 = 20 \text{ cm} $$

Answer: The diameter of the circle is 20 cm.

This simple formula is useful for many exam questions on circles.

Summary Table: Circle Elements and Their Definitions

Here's a quick comparison of the main elements of a circle:

Use this table to quickly recall definitions and properties during your exams.