What is Circles Class 10: Complete Guide for NCERT Maths Chapter

Definition and Basic Terms of Circles

A circle is the set of all points in a plane that are at a fixed distance from a fixed point called the center. This fixed distance is called the radius.

Key terms to remember:

• Radius (r): Distance from the center to any point on the circle.
• Diameter (d): A chord passing through the center; $d = 2r$.
• Chord: A line segment joining two points on the circle.
• Arc: A part of the circumference between two points.
• Circumference: The complete boundary of the circle.
• Sector: A region bounded by two radii and the arc between them.
• Tangent: A line that touches the circle at exactly one point.

Understanding these terms is crucial for solving problems in the Circles chapter.

Properties of Chords and Their Importance

Chords have several important properties:

• The perpendicular from the center to a chord bisects the chord.
• Equal chords are equidistant from the center.
• The diameter is the longest chord of a circle.

Example: If a chord of length 8 cm is 3 cm away from the center, find the radius.

Using the right triangle formed:

$$r^2 = 3^2 + 4^2 = 9 + 16 = 25$$

So, $r = 5$ cm.

These properties help in solving many geometry problems involving circles.

Understanding Tangents and Their Properties

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

Key properties:

• A tangent is perpendicular to the radius at the point of contact.
• From an external point, two tangents to a circle are equal in length.

Formula: If $PT$ is a tangent from point $P$ to the circle with center $O$ and radius $r$, then:

$$OP ot PT$$

This means $OP$ is perpendicular to the tangent at $T$.

Understanding tangents is important for solving problems related to circles and lines.

Formulas for Circumference and Area of a Circle

Two important formulas every student must remember:

• Circumference (perimeter) of a circle:

$$C = 2\pi r$$

• Area of a circle:

$$A = \pi r^2$$

Where $r$ is the radius and $\pi$ (pi) is approximately 3.1416.

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

• Circumference: $2 \times 3.1416 \times 7 = 43.9824$ cm
• Area: $3.1416 \times 7^2 = 153.9384$ cm²

These formulas are frequently asked in Class 10 exams.

Angles in a Circle: Key Theorems and Applications

Several angle properties help solve circle problems:

• The angle subtended by a diameter at the circumference is a right angle (90°).
• Angles subtended by the same arc at the circumference are equal.
• The angle between a tangent and a chord is equal to the angle subtended by the chord in the alternate segment.

These theorems are used to find unknown angles and prove geometric results.

Example: If $AB$ is a diameter and $C$ is a point on the circle, then $\angle ACB = 90^\circ$.

These properties are part of the NCERT Class 10 syllabus and important for exams.

Comparison Table: Radius, Diameter, and Chord

This table helps clarify the differences and relations among these terms.