Class 9 Maths - RAJASTHAN
Circles
The Chapter 'Circles' in Class 9 Mathematics under the Rajasthan Board (RBSE) introduces students to the fundamental geometric properties of circles. A circle is the locus of a point equidistant from a fixed point. This chapter covers essential terms like chords, arcs, sectors, and segments. Students learn crucial theorems regarding the perpendicular from the centre to a chord, equal chords and their distances from the centre, angles subtended by chords and arcs at the centre and circumference, cyclic quadrilaterals, and angle properties of cyclic quadrilaterals. Mastering these concepts builds a strong foundation for advanced geometry in higher classes and helps score high marks in board-pattern exams.
Start Learning FreeKey Concepts
Chord and Centre Perpendicular Theorem
The perpendicular drawn from the centre of a circle to a chord bisects the chord, and conversely, the line joining the centre to the midpoint of a chord is perpendicular to the chord.
Equal Chords and Distance
Equal chords of a circle are equidistant from the centre, and chords that are equidistant from the centre of a circle are equal in length.
Angle Subtended by an Arc
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.
Angles in the Same Segment
Angles in the same segment of a circle are equal, and the angle in a semicircle is a right angle (90 degrees).
Cyclic Quadrilateral
A quadrilateral is called cyclic if all four of its vertices lie on a circle, and the sum of either pair of opposite angles of a cyclic quadrilateral is 180 degrees.
Important Formulas
Board Exam Info
In the Rajasthan Board (RBSE) Class 9 Mathematics examination, the chapter 'Circles' typically carries around 6 to 8 marks. Questions often include 1-mark multiple-choice questions, 2-mark short answer questions based on direct theorem applications, and 4-mark proof-based questions or numerical problems involving cyclic quadrilaterals and angles.
Frequently Asked Questions
What is the difference between a chord and a secant?
A chord is a line segment whose endpoints both lie on the circle, whereas a secant is a straight line that intersects the circle at two distinct points and extends indefinitely in both directions.
How do we prove that a quadrilateral is cyclic?
You can prove a quadrilateral is cyclic by showing that the sum of either pair of its opposite angles is supplementary (180 degrees), or by showing that an exterior angle is equal to the interior opposite angle.
Is the angle in a semicircle always 90 degrees?
Yes, because the diameter subtends a straight angle (180 degrees) at the centre, and half of that at the circumference is 90 degrees.
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