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I'm Up and Down, and Round and Round
This chapter builds the idea of a circle from its centre, radius, diameter, chords, arcs and angles. For CBSE-aligned preparation, students should connect each result with a clear diagram because most circle questions become simple after marking the centre, equal radii, chord, and required angles. The main exam value of this chapter is theorem use: perpendicular from centre to chord, equal chords, angle at centre and circumference, angles in the same segment, and cyclic quadrilateral properties. A good answer usually needs a labelled figure, the correct theorem statement, and one or two logical steps.
Difficulty
Medium
Study time
64-80 min
Plan by time
Pick the window that matches what you have right now.
If you have 15 min
Last-pass revision
Skim the Quick Revision table — definitions, formulas, and the traps board examiners reuse.
Open Quick RevisionIf you have 45 min
Targeted practice
Read the high-priority concepts, then take the chapter MCQ quiz to find weak spots.
Start MCQ QuizIf you have 64 min
First full pass
Walk every concept in chapter order, then revise and quiz. Best for the first time you study this chapter.
Open Key ConceptsChapter Learning Map
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Key Concepts
Concepts grouped the way the chapter is taught — open the bucket that matches what you want to revise.
Core Concepts
high priorityOpen the chapter concepts in a clean revision order.
Definition of a Circle
A circle is the set of all points in a plane that are at the same distance from a fixed point called the centre. This fixed distance is called the radius.
Symmetries of a Circle
A circle has reflection symmetry about every diameter and rotational symmetry about its centre for any angle of rotation.
Chord and Its Properties
A chord is a line segment joining two points on a circle. The perpendicular from the centre of a circle to a chord bisects the chord, and equal chords are equidistant from the centre.
Three Points Determine a Circle
One and only one circle can pass through three non-collinear points. Its centre is found at the intersection of perpendicular bisectors of two joining segments.
Angle Subtended by Arc/Chord at Centre vs at Circumference
The angle subtended by an arc or chord at the centre of a circle is twice the angle subtended by it at any point on the remaining part of the circle.
Angles in Same Segment
Angles in the same segment of a circle are equal. These angles stand on the same chord and have their vertices on the same arc segment.
Cyclic Quadrilateral
A cyclic quadrilateral is a quadrilateral whose four vertices lie on one circle. In a cyclic quadrilateral, each pair of opposite angles is supplementary.
Concyclic Points
Points are called concyclic if they lie on the same circle. Four points can often be tested for concyclicity by checking whether a pair of opposite angles is supplementary.
Exam Intelligence
Use this section to decide what deserves the most revision time.
High Probability Topics
- Definition of a Circle
- Symmetries of a Circle
- Chord and Its Properties
- Three Points Determine a Circle
- Angle Subtended by Arc/Chord at Centre vs at Circumference
- Angles in Same Segment
- Cyclic Quadrilateral
- Concyclic Points
Common Traps
- Calling every chord a diameter; a diameter must pass through the centre.
- Using chord bisection without checking the perpendicular from the centre.
- Forgetting the non-collinear condition for a unique circle through three points.
- Reversing the centre-circumference angle relation.
- Applying same segment angle equality when vertices are on different segments.
- Adding adjacent angles instead of opposite angles in cyclic quadrilateral questions.
Likely Question Types
- MCQ: concept checks, applications, and common mistakes
- Very short answer: definitions, formulas, or conditions
- Short answer: worked method, example, or reason-based explanation
- Case-based: chapter scenario with concept-linked subparts
Quick Revision
Concept, formula or equation to remember, and the trap that loses marks — in one scannable view.
- A circle is built from one fixed centre and equal radius to every boundary point.
- Diameter is the longest chord and equals twice the radius.
- Chord theorems connect perpendicular distance from centre with bisection and equality of chords.
- Three non-collinear points determine exactly one circle through perpendicular bisectors.
- The angle at the centre is double the angle at the circumference on the same chord or arc.
- Angles in the same segment are equal, while opposite angles of a cyclic quadrilateral add to 180 degrees.
- Concyclic point questions often reduce to proving one opposite angle pair is supplementary.
- Definition of a Circle: A circle is the set of all points in a plane that are at the same distance from a fixed point called the centre. This fixed distance is cal…
Practice
Use short concept checks first, then move into the full chapter test.
Free Chapter MCQ Quiz
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