What is Circles Class 9: Complete Guide for NCERT Students

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. This fixed point is called the centre of the circle, and the fixed distance is called the radius.

Key elements of a circle:

• Centre (O): The fixed point inside the circle.
• Radius (r): The distance from the centre to any point on the circle.
• Diameter (d): A line segment passing through the centre with endpoints on the circle. It is twice the radius: $d = 2r$.
• Chord: A line segment with both endpoints on the circle but not passing through the centre.
• Arc: A part of the circumference of a circle.
• Circumference: The total distance around the circle.

Understanding these elements is fundamental to solving problems related to circles in Class 9 NCERT Mathematics.

Properties of Circles Every Class 9 Student Should Know

Circles have several important properties that help solve geometry problems:

• All radii of a circle are equal.
• The diameter is the longest chord of the circle.
• A chord equidistant from the centre is equal in length to another chord at the same distance.
• The perpendicular from the centre to a chord bisects the chord.
• The tangent to a circle is perpendicular to the radius at the point of contact.

These properties form the basis for many theorems and proofs in the NCERT Class 9 chapter on circles.

Formulas Related to Circles in Class 9 NCERT

Knowing the right formulas is crucial for solving numerical problems on circles.

Here, $r$ is the radius and $\pi$ is approximately 3.1416.

Worked Example:

Find the circumference and area of a circle with radius 7 cm.

• Circumference = $2 \times \pi \times 7 = 44$ cm (approx)
• Area = $\pi \times 7^2 = 154$ cm² (approx)

These formulas are frequently tested in Class 9 exams.

Tangent to a Circle: Definition and Important Facts

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

Important facts about tangents:

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

These facts are vital when solving problems involving tangents in Class 9 NCERT Maths.

Difference Between Radius, Diameter, and Chord

Understanding the difference between radius, diameter, and chord helps clarify circle geometry:

This comparison aids in visualising circle parts and solving related problems.

Common Theorems on Circles for Class 9 Exams

Several theorems on circles are important for Class 9 students:

• Theorem 1: Equal chords of a circle are equidistant from the centre.
• Theorem 2: The perpendicular from the centre to a chord bisects the chord.
• Theorem 3: The angle subtended by a diameter at the circumference is a right angle.

Example:

If a chord is 24 cm long and is 7 cm from the centre, find the radius.

Using Pythagoras theorem in the right triangle formed:

$$ r^2 = 7^2 + 12^2 = 49 + 144 = 193 $$

So, radius $r = \sqrt{193} \approx 13.89$ cm.

These theorems and examples are essential for NCERT Class 9 exams.