Class 9 Maths - UP
Circles
The chapter 'Circles' in Class 9 Mathematics under the UPMSP curriculum builds a strong foundation in geometry by exploring the properties of circles, chords, arcs, and angles subtended by them. Students learn essential geometric theorems regarding perpendiculars from the centre to a chord, equal chords and their distances from the centre, and cyclic quadrilaterals. This chapter is vital for board exams as it tests logical reasoning and theorem application. Scoring well here requires clear understanding of diagrams and step-by-step proof writing, making it a high-scoring area in geometry.
Start Learning FreeKey Concepts
Chord and Arc Properties
A line segment joining any two points on a circle is called a chord. Equal chords of a circle subtend equal angles at the centre and are at equal distances from the centre.
Perpendicular from Centre to Chord
The perpendicular drawn from the centre of a circle to a chord bisects the chord. Conversely, the line drawn through the centre to bisect a chord is perpendicular to the chord.
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. Also, angles in the same segment of a circle are equal.
Cyclic Quadrilateral
A quadrilateral is called cyclic if all its four vertices lie on a circle. The sum of either pair of opposite angles of a cyclic quadrilateral is 180 degrees.
Circle through Three Points
There is one and only one circle passing through three given non-collinear points, which helps in determining a unique circle using a triangle's vertices.
Important Formulas
Board Exam Info
In the Uttar Pradesh (UPMSP) Class 9 Mathematics examination, geometry chapters like Circles typically carry around 8 to 12 marks. Common question types include short-answer questions based on finding unknown angles, and long-answer proof-based questions involving cyclic quadrilaterals or chord properties.
Frequently Asked Questions
What is the difference between a chord and a diameter?
A chord is any line segment joining two points on a circle, while a diameter is a special chord that passes through the centre of the circle and is the longest chord.
Are proofs of theorems asked in the UPMSP Class 9 exam?
Yes, direct proofs of important theorems, such as the angle subtended by an arc at the centre theorem or cyclic quadrilateral properties, are frequently asked as long-answer questions.
How do I prove a quadrilateral is cyclic?
You can prove a quadrilateral is cyclic by showing that the sum of either pair of opposite angles is 180 degrees, or by showing that an exterior angle is equal to the interior opposite angle.
Learn Circles with Your AI Tutor
10 different ways to study this chapter. Free for 3 chapters per day.
Lecture
Key Points
Interactive
Quiz
Flashcards