Class 9 Mathematic Circle All Formula & Concept Notes | Formula Sheet

Here’s a comprehensive overview of the key concepts and formulas related to circles in Class 9 Mathematics:

1. Definitions and Basic Concepts

• Circle: A set of all points in a plane that are at a fixed distance (radius) from a fixed point (center).
• Radius: The distance from the center of the circle to any point on the circle.
• Diameter: The longest chord of the circle, passing through the center. It is twice the radius.
• Chord: A line segment with both endpoints on the circle.
• Secant: A line that intersects the circle at two points.
• Tangent: A line that touches the circle at exactly one point.
• Arc: A part of the circumference of the circle.
• Sector: The region enclosed by two radii and the arc between them.
• Segment: The region enclosed by a chord and the arc between its endpoints.

2. Angles Related to Circles

• Central Angle: An angle whose vertex is the center of the circle and whose sides are radii of the circle. The measure of a central angle is equal to the measure of the arc it intercepts.

• Inscribed Angle: An angle whose vertex is on the circle and whose sides intersect the circle. The measure of an inscribed angle is half the measure of the arc it intercepts.

• Angle in the Semicircle: An angle inscribed in a semicircle is a right angle (90 degrees).

• Angles in the Same Segment: Angles in the same segment of a circle are equal.

• Opposite Angles of a Cyclic Quadrilateral: The sum of the opposite angles of a cyclic quadrilateral is 180 degrees.

3. Length of an Arc

The length of an arc $l$ of a circle can be calculated using the formula: $l = \frac{\theta}{360} \times 2 \pi r$ where:

• $\theta$ is the angle subtended by the arc at the center of the circle.
• $r$ is the radius of the circle.
• $\pi \approx 3.14$

4. Area of a Sector

The area $A$ of a sector of a circle can be calculated using: $A = \frac{\theta}{360} \times \pi r^2$ where:

• $\theta$ is the angle subtended by the sector at the center of the circle.
• $r$ is the radius of the circle.

5. Area of a Segment

The area of a segment of a circle (the area between a chord and the arc) can be found by subtracting the area of the triangular portion from the area of the sector: $\text{Area of Segment} = \text{Area of Sector} - \text{Area of Triangle}$

6. Length of Chord

The length $c$ of a chord can be calculated using: $c = 2 r \sin \left(\frac{\theta}{2}\right)$ where:

• $\theta$ is the angle subtended by the chord at the center of the circle.
• $r$ is the radius of the circle.

7. Power of a Point

For a point $P$ outside a circle, if two tangents are drawn from $P$ to the circle, then the lengths of these tangents are equal. If $PT$ and $PT'$ are the tangents drawn from $P$ to the circle, and $PC$ is a secant intersecting the circle at points $A$ and $B$, then: $PT^2 = PA \cdot PB$

8. Equation of a Circle

In coordinate geometry, the standard equation of a circle with center $(h, k)$ and radius $r$ is: $(x - h)^2 + (y - k)^2 = r^2$

If the center is at the origin (0,0), the equation simplifies to: $x^2 + y^2 = r^2$

9. Chord Properties

• Perpendicular from Center: The perpendicular drawn from the center of a circle to a chord bisects the chord.
• Equal Chords: Chords equidistant from the center are equal in length.

These are the fundamental concepts and formulas related to circles that you should be familiar with in Class 9 Mathematics. If you need more details or have specific questions about any of these topics,