Quick Revision Notes: Class 9 Mathematics

Chapter: Circles (वृत्त)

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### 1. Important Definitions (महत्वपूर्ण परिभाषाएँ)

• Circle (वृत्त): The collection of all points in a plane which are at a fixed distance (radius) from a fixed point (centre).
• Centre (केंद्र): The fixed point inside a circle from which all points on the boundary are equidistant.
• Radius (त्रिज्या - $r$): The line segment joining the centre of the circle to any point on its boundary.
• Chord (जीवा): A line segment joining any two points on the circle.
• Diameter (व्यास - $d$): A chord that passes through the centre of the circle. It is the longest chord and $d = 2r$.
• Secant (छेदक रेखा): A line that intersects a circle at two distinct points.
• Tangent (स्पर्श रेखा): A line that touches the circle at only one point.
• Arc (चाप): A continuous piece of a circle.
  • Minor Arc (लघु चाप): Less than half of the circle.
  • Major Arc (दीर्घ चाप): Greater than half of the circle.
• Circumference (परिधि): The perimeter or total length of the boundary of a circle ($2\pi r$).
• Segment (वृत्तखंड): The region between a chord and either of its arcs.
  • Minor Segment (लघु वृत्तखंड) and Major Segment (दीर्घ वृत्तखंड).
• Sector (त्रिज्यखंड): The region between two radii and the enclosing arc.
  • Minor Sector (लघु त्रिज्यखंड) and Major Sector (दीर्घ त्रिज्यखंड).
• Concentric Circles (संकेन्द्रिक वृत्त): Circles with the same centre but different radii.
• Cyclic Quadrilateral (चक्रीय चतुर्भुज): A quadrilateral whose all four vertices lie on the circumference of a circle.

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### 2. Key Theorems & Properties (महत्वपूर्ण प्रमेय और गुण)

Theorem 1: Perpendicular from Centre to Chord

• Statement: The perpendicular drawn from the centre of a circle to a chord bisects the chord.
• Inverse: The line drawn through the centre of a circle to bisect a chord is perpendicular to the chord.
• If $OM \perp AB$, then $AM = MB$.

Theorem 2: Equal Chords and Distance from Centre

• Statement: Equal chords of a circle are equidistant from the centre.
• Inverse: Chords equidistant from the centre of a circle are equal in length.
• If $AB = CD$, then $OM = ON$ (where $OM$ and $ON$ are perpendicular distances).

Theorem 3: Angle Subtended by an Arc at the Centre

• Statement: The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.
• $\angle AOB = 2 \angle APB$

Theorem 4: Angles in the Same Segment

• Statement: Angles in the same segment of a circle are equal.
• $\angle ACB = \angle ADB$ (Both lie in the same segment).

Theorem 5: Angle in a Semicircle

• Statement: The angle subtended by a diameter at any point on the semicircle is a right angle ($90^\circ$).
• $\angle APB = 90^\circ$ (if $AB$ is a diameter).

Theorem 6: Equal Chords and Subtended Angles

• Statement: Equal chords of a circle subtend equal angles at the centre.
• Inverse: If the angles subtended by the chords at the centre of a circle are equal, then the chords are equal.
• If $AB = CD$, then $\angle AOB = \angle COD$.

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### 3. Cyclic Quadrilaterals (चक्रीय चतुर्भुज)

• Definition: A quadrilateral $ABCD$ is called cyclic if all the four vertices lie on a circle.
• Theorem (Sum of Opposite Angles): The sum of either pair of opposite angles of a cyclic quadrilateral is $180^\circ$ (supplementary).
  • $\angle A + \angle C = 180^\circ$
  • $\angle B + \angle D = 180^\circ$
• Exterior Angle Property: If a side of a cyclic quadrilateral is produced, then the exterior angle is equal to the interior opposite angle.

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### 4. Quick Formula Summary (सूत्र सारांश)

• Diameter ($d$): $d = 2 \times r$ (where $r$ is radius)
• Radius ($r$): $r = \frac{d}{2}$
• Circumference of a Circle: $C = 2\pi r$ (or $\pi d$)
• Area of a Circle: $A = \pi r^2$
• Area of a Semicircle: $\frac{1}{2} \pi r^2$
• Length of an Arc ($l$): $l = \frac{\theta}{360^\circ} \times 2\pi r$ (where $\theta$ is the angle of the sector)
• Area of a Sector: $\text{Area} = \frac{\theta}{360^\circ} \times \pi r^2$

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Tip for MP Board Exam: Always draw neat and labeled diagrams for geometry questions. Write "Given", "To Prove", and "Proof" systematically to secure full marks.