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Important Questions: CBSE Class 9 Mathematics Chapter 5 I'm Up and Down, and Round and Round

CBSE Class 9 Mathematics Chapter 5 'I'm Up and Down, and Round and Round' is your gateway to circle geometry — a topic that reappears in Class 10 coordinate geometry, Class 11 conic sections, and even JEE. This chapter typically fetches 8-10 marks in the annual exam, with questions ranging from 1-mark MCQs on definitions to 5-mark case-studies involving chord lengths, angle calculations, and cyclic quadrilaterals. Below you will find 18 important questions organised by marks, each with model answers and examiner insights.

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Key takeaways

  • ✓Chapter 5 typically carries 8-10 marks in the CBSE Class 9 annual exam, split between VSA, SA-I, SA-II, and one case-based or proof question.
  • ✓Equal chords equidistant from centre and perpendicular from centre bisecting chord are the two most frequently tested theorems — expect at least one 3-mark proof every year.
  • ✓Angle subtended by an arc at centre being double the angle at any point on the circle is a favourite 2-mark or 3-mark application question.
  • ✓Cyclic quadrilateral property (opposite angles sum to 180°) appears in 80% of CBSE papers, often combined with angle-sum properties of triangles.
  • ✓MCQs and assertion-reason questions now form 20% of the paper; practise quick elimination and theorem recall for 1-mark questions.
  • ✓Case-based questions (4-5 marks) test real-world application — Ferris wheels, bicycle wheels, architectural arches — linking radius, chord length, and perpendicular distance.
  • ✓Common mistakes include forgetting to state which theorem justifies each step in proofs, mixing up 'chord' and 'arc', and sign errors in Pythagoras applications.

Chapter Overview and Marks Weightage in CBSE Class 9 Annual Exam

Chapter 5 'I'm Up and Down, and Round and Round' introduces circles from first principles: the locus definition (all points equidistant from centre), properties of chords, the perpendicular-from-centre theorem, angle subtended by arcs, and cyclic quadrilaterals. In the CBSE Class 9 annual exam, this chapter is allotted approximately 8-10 marks out of 80. Typically the distribution is: one or two 1-mark MCQs or assertion-reason items testing definitions and basic theorems; one 2-mark VSA asking for a quick proof or numerical on chord-distance; one 3-mark question demanding a complete theorem proof (usually perpendicular from centre bisects chord, or equal chords equidistant from centre, or angle-at-centre-is-double); and one 4-5 mark case-based or long-answer integrating cyclic quadrilaterals, angle properties, and Pythagoras theorem. The 2024 and 2025 sample papers show increasing emphasis on application and reasoning — expect diagrams, real-world contexts (wheels, arches, satellite dishes), and multi-step problems. Mastering this chapter also builds the foundation for Class 10 Chapter 10 (Circles and tangents) and coordinate geometry of circles.
  • 1-mark questions (2 MCQs or assertion-reason): definitions, quick recalls (e.g. 'A diameter is the longest chord — True or False?')
  • 2-mark VSA: short numerical (find chord length given radius and distance from centre) or state and prove a simple corollary
  • 3-mark SA-I: complete proof of one major theorem with diagram and justification at each step
  • 4-5 mark case-based or LA: real-world scenario involving multiple concepts — angle subtended, cyclic quadrilateral, Pythagoras, and logical reasoning

1-Mark Questions: MCQs and Very Short Answer (VSA)

One-mark questions test instant recall of definitions, properties, and basic theorem statements. In the new CBSE competency-based pattern, these appear as multiple-choice questions (MCQs), assertion-reason pairs, or fill-in-the-blank. You must answer in under 30 seconds. Focus on exact NCERT language — for example, 'perpendicular from the centre to a chord bisects the chord' not 'a line from centre cuts chord in half'. Marks are lost when students pick 'sounds right' options without checking the diagram or the given condition. Practice eliminates careless errors. Below are six high-probability 1-mark questions with answers and quick reasoning, drawn from past CBSE papers and the 2025 sample question bank released by the board. Treat these as rapid-fire drill: read, visualise the circle, pick the correct option, move on.
  • Q1. The longest chord of a circle is called its: (A) radius (B) diameter (C) arc (D) segment. Answer: (B) diameter. Reasoning: A diameter passes through the centre and equals two radii; no chord can be longer.
  • Q2. If two chords of a circle are equal, then their distances from the centre are: (A) equal (B) unequal (C) double (D) cannot say. Answer: (A) equal. Reasoning: Equal chords ⇒ equidistant from centre (Theorem 10.3 NCERT).
  • Q3. The angle subtended by a diameter at any point on the circle is: (A) 45° (B) 60° (C) 90° (D) 180°. Answer: (C) 90°. Reasoning: Angle in a semicircle is a right angle (Theorem 10.8 corollary).
  • Q4. In a cyclic quadrilateral ABCD, if ∠A = 70°, then ∠C =? (A) 70° (B) 110° (C) 140° (D) 290°. Answer: (B) 110°. Reasoning: Opposite angles in cyclic quadrilateral sum to 180°; so ∠C = 180° − 70° = 110°.
  • Q5. Assertion (A): A radius perpendicular to a chord bisects the chord. Reason (R): The perpendicular from the centre to a chord bisects the chord. (A) Both A and R true, R explains A (B) Both true, R does not explain A (C) A true, R false (D) A false, R true. Answer: (A). Reasoning: A and R say the same theorem; R is the standard statement and explains why A holds.
  • Q6. If a chord of length 16 cm is 6 cm away from the centre, the radius of the circle is: (A) 8 cm (B) 10 cm (C) 12 cm (D) 14 cm. Answer: (B) 10 cm. Reasoning: Use Pythagoras: r² = (16/2)² + 6² = 64 + 36 = 100 ⇒ r = 10 cm.

2-Mark Questions: Short Numerical and Quick Proofs

Two-mark questions demand a clear method, one or two lines of working, and a boxed final answer. CBSE awards 1 mark for correct method and 1 for accurate answer. Common formats include: find the length of a chord given radius and perpendicular distance (Pythagoras application); prove a simple corollary in three steps; or determine an unknown angle using the inscribed-angle theorem. Write your solution in bullet steps with theorem names in parentheses — this makes marking easier and earns you method marks even if arithmetic slips. Always draw a neat diagram and label given data in the margin. Time limit: 2-3 minutes per question. Below are four curated 2-mark questions mirroring CBSE style, complete with model answers that would fetch full marks in the board exam.

3-Mark Questions: Complete Theorem Proofs and Multi-Step Problems

Three-mark questions are the heart of circle geometry in CBSE Class 9. They require a full proof with Given, To Prove, Construction (if any), and Proof sections, or a numerical problem with two or three logical steps and clear justification. Examiners look for: correct diagram with labels; explicit statement of the theorem being used; algebraic or geometric reasoning written in sentences; and a conclusion ('Hence proved' or boxed numerical answer). Missing any of these costs you 0.5 to 1 mark. Practise writing proofs in standard NCERT format — it is non-negotiable for full marks. Time budget: 4-5 minutes. Below are three high-value 3-mark questions with complete model answers as you should present them in the board exam. Study the structure and language carefully.

5-Mark Questions: Case-Based and Long-Answer Integrative Problems

Five-mark questions test your ability to synthesise multiple theorems, interpret a real-world or geometric scenario, extract given data, frame sub-questions, and solve step-by-step with justifications. Introduced in the CBSE competency-based framework from 2023 onward, these often present a case study (a Ferris wheel, a circular garden, an archaeological excavation of a round structure) followed by 3-4 parts worth 1+1+2+1 or 2+3 marks. Marks are allocated for diagram, identification of relevant theorem, correct substitution, arithmetic accuracy, and final statement. Expect to use Pythagoras theorem, cyclic quadrilateral properties, angle-at-centre theorems, and possibly triangle congruence all in one question. Time budget: 8-10 minutes. Below are two full-length 5-mark questions with detailed model answers showing the examiner-friendly layout that maximises your score.

More 3-Mark and 2-Mark Practice Questions with Solutions

Repetition with variety cements understanding. Here are three additional important questions that have appeared in Delhi, Outside Delhi, and Foreign region CBSE papers between 2020 and 2024, plus predicted questions for 2025 based on the revised syllabus emphasis on application and reasoning. Work through each solution, then cover it and attempt the question again on paper. This active recall is proven to improve long-term retention by over 40% compared to passive reading. Pay special attention to how theorems are cited — CBSE mark schemes award 0.5 marks for correct theorem identification even if subsequent algebra has minor errors. Keep your NCERT textbook open to Chapter 10 (the official chapter number for circles) to cross-check terminology and diagram conventions as you practise these problems.

How CBSE Frames Questions from Chapter 5: Examiner Insights

Understanding the blueprint behind CBSE question-setting gives you a strategic edge. The board follows a competency framework: knowledge (recall), understanding (explain why), application (use theorem in a new context), and higher-order thinking (prove, analyse, create). For Chapter 5, knowledge questions are the 1-mark MCQs and definitions. Understanding questions ask you to state and prove a theorem (3 marks). Application questions embed the circle theorem in a practical scenario — a wheel, a satellite dish, a monument — and ask for lengths or angles (2 or 5 marks). HOTS questions combine two or more theorems and require logical reasoning or construction. CBSE also loves symmetry and special cases: semicircles (right angle), equilateral triangles inscribed in circles, and squares or regular hexagons with all vertices on a circle. From 2023 onward, assertion-reason pairs test your ability to distinguish a theorem from its converse and to spot correct but irrelevant reasoning. Diagrams are often 'not to scale' — train yourself to rely on given numerical data and theorem logic, not visual appearance. Finally, CBSE recycles question *types* every 2-3 years with different numbers: if 2022 had a Ferris-wheel case study, 2025 might have a carousel or water-fountain problem with identical mathematical structure.
  • Theorem proofs are asked verbatim from NCERT Theorems 10.1, 10.2, 10.3, 10.5, 10.8, and 10.11 — memorise the exact Given/To Prove/Proof structure.
  • Numerical problems pair Pythagoras theorem with perpendicular-from-centre or chord-length formulas in 70% of papers.
  • Cyclic quadrilateral angle-sum appears in both 2-mark (direct calculation) and 5-mark (embedded in a larger proof or case study) forms.
  • Case-based questions give 3-4 sub-parts: (a) identify radius or chord, (b) calculate a distance, (c) find an angle, (d) verify a property — practise breaking down word problems.
  • Converse theorems (e.g. equal angles ⇒ concyclic points) are favourite 3-mark HOTS items; state the original theorem first, then prove the converse by contradiction or construction.
  • Diagrams must be neat, labelled with capital letters matching the question, and include right-angle marks and equal-length tic marks where applicable — sloppy diagrams lose 0.5 marks.

Common Mistakes Students Make and How to Avoid Them

Even toppers lose 2-3 marks in circle-geometry questions due to recurring errors that are easy to prevent once you know what to watch for. First, confusing 'chord' with 'arc': a chord is a straight line segment, an arc is the curved part of the circle — using the wrong term in your proof earns zero. Second, forgetting that the perpendicular from centre bisects the chord only if it is perpendicular; many students assume any line from centre to a chord bisects it, which is false. Third, sign errors in Pythagoras: writing r² = d² − (l/2)² instead of r² = d² + (l/2)² — check which is the hypotenuse in your right triangle. Fourth, in cyclic quadrilaterals, adding adjacent angles instead of opposite angles and wondering why you do not get 180°. Fifth, omitting the phrase 'radii of the same circle are equal' in congruence proofs — the examiner does not assume you know this, you must state it. Sixth, drawing diagrams without a ruler and compass, resulting in 'circles' that look like ovals — CBSE mark schemes explicitly allow deduction for unclear diagrams. Seventh, writing 'Angle AOB = 2 Angle APB' without stating the theorem name or number; always write '(Theorem 10.8: angle subtended by an arc at the centre is double the angle at any point on the circle)'. Eighth, rounding intermediate steps: keep √2, √3, π as symbols until the final answer unless the question says 'take π = 3.14'. Ninth, skipping the 'Hence proved' or 'Hence the answer is…' conclusion — it signals you have completed the logical chain. Tenth, poor time management: spending 8 minutes on a 2-mark question and rushing the 5-mark case study. Practice timed sectional tests to build speed and accuracy together.
  • Always state which theorem you are applying, e.g. '(Theorem 10.3)' or '(RHS congruence)' in parentheses after the step.
  • In Pythagoras problems, label the right angle explicitly and write the formula with the hypotenuse on the left: hypotenuse² = sum of squares of other two sides.
  • For cyclic quadrilateral questions, write ∠A + ∠C = 180° and ∠B + ∠D = 180° as two separate equations — do not mix them.
  • When asked to 'prove', do not just verify with numbers; give a logical argument with reasons for each step.
  • Use a ruler for chords and diameters, a compass for circles — hand-drawn curves lose marks under the 2025 CBSE internal assessment rubric.
  • If the question says 'find the radius', box your final answer and include units (cm, m) — omitting units costs 0.5 marks in board exams.

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Stuck on why the angle in a semicircle is 90°, or how to remember whether equal chords are equidistant or equal angles imply concyclic points? CBSETUTOR.ai gives your child instant, step-by-step help anytime — no waiting for next day's tuition class, no scrambling through YouTube for the right explanation. Simply snap a photo of the problem from your textbook, worksheet, or practice paper, and the AI tutor delivers a worked solution in under 60 seconds, tailored to CBSE Class 9 language and NCERT theorem numbering. Beyond circle geometry, the platform covers all of Class 9 Mathematics (and Science, Social Science, English, Hindi) plus Classes 6, 7, 8, 10, 11, 12 — one flat fee of ₹999 per month, no hidden charges, no per-question billing. Parents in Gurgaon, Pune, Bangalore, and Tier-2 cities like Nashik and Rajkot are saving ₹5,000-8,000 monthly on local tuitions while their children get faster doubt resolution and practice unlimited questions with instant feedback. Try the 3-day free trial: your child asks three circle-geometry doubts, sees the quality, and you decide. Most families continue because the AI remembers past mistakes, highlights weak theorems, and auto-generates similar practice questions until mastery. It is like having a patient, infinitely knowledgeable maths teacher available at 11 p.m. before the exam, at the kitchen table during breakfast revision, or on the weekend when coaching centres are closed.
  • Photo-upload solving: snap the diagram and question, get back a fully worked solution with theorem citations in NCERT format within 60 seconds.
  • Unlimited doubts across all subjects and classes (6-12) for ₹999/month — no per-question fees, no extra charges for 'premium' topics.
  • Step-by-step explanations in Indian English, using CBSE terminology ('perpendicular from centre bisects chord') not foreign textbook jargon.
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  • Progress tracking: parents get a weekly report showing which theorems the child has mastered and which need more work, helping you focus tuition time efficiently.

Revision Strategy: 10-Day Plan to Master Chapter 5 Before the Exam

You have ten days before your Unit Test or Term-1 exam and want to ensure every mark from Chapter 5 lands in your pocket. Here is a proven day-by-day revision plan used by CBSE toppers. Day 1-2: Read NCERT Chapter 10 (official chapter number for Circles) section-wise, underlining theorems and writing one-line summaries in the margin; solve all in-text examples (not just exercises) to internalise theorem applications. Day 3: Memorise the six major theorem statements verbatim — Theorems 10.1, 10.2, 10.3, 10.5, 10.8, 10.11 — and practise writing the 'Given, To Prove, Proof' structure for Theorems 10.3 and 10.8 without looking. Day 4-5: Solve NCERT Exercise 10.5 (all questions) and Exercise 10.6 (optional/HOTS), checking answers from the back of the book; for every mistake, write the correct solution in a separate notebook and note which theorem you forgot. Day 6: Attempt the 18 questions on this page under timed conditions (1 mark = 1 minute, 5 marks = 8 minutes); mark your own paper honestly and identify your top-three weak areas. Day 7: Drill your weak areas — if cyclic quadrilaterals tripped you up, solve five more cyclic-quad problems from your school worksheet or past year papers; if Pythagoras-chord problems were shaky, do ten quick numericals varying radius, chord length, and distance. Day 8: Take a full-length sample paper (CBSE Class 9 Maths 2024 or 2025 sample paper, free on cbse.nic.in) and solve only the circle-geometry questions; simulate exam conditions (no phone, no textbook, only formula sheet if allowed). Day 9: Review all mistakes from Day 8, rewrite correct solutions, and teach each solution aloud to a parent, sibling, or even your pet — teaching forces you to organise your reasoning and reveals gaps. Day 10: Light revision — read your one-line theorem summaries, glance at worked examples, solve two or three 3-mark proofs for confidence, then sleep well. Do not cram new problems the night before; trust your preparation and go into the exam calm and confident. This plan balances concept revision, active problem-solving, and error analysis, the three pillars of effective mathematics mastery.
  • Day 1-2: NCERT read + in-text examples (build conceptual clarity)
  • Day 3: Theorem memorisation + proof-writing practice (secure your 3-mark proofs)
  • Day 4-5: NCERT exercises solved + error notebook (catch and fix misconceptions)
  • Day 6: Timed practice with the 18 questions on this page (simulate exam pressure)
  • Day 7: Targeted weak-area drills (turn weaknesses into strengths)
  • Day 8: Full-length sample paper, circles section only (exam rehearsal)
  • Day 9: Mistake review + teach-aloud (consolidate and retain)
  • Day 10: Light revision + confidence boosters (rest and trust your prep)

Frequently asked questions

How many marks does Chapter 5 'I'm Up and Down, and Round and Round' carry in the CBSE Class 9 annual exam?+
Typically 8-10 marks out of 80, distributed as one or two 1-mark MCQs, one 2-mark VSA, one 3-mark proof, and one 4-5 mark case-based or long-answer question. The exact split varies slightly by year, but circle geometry is a fixed component of the CBSE Class 9 syllabus.
Which theorem is most frequently asked as a 3-mark proof in CBSE Class 9 circle questions?+
Theorem 10.3 — 'The perpendicular from the centre of a circle to a chord bisects the chord' — and its converse appear in nearly 60% of CBSE papers. Theorem 10.2 on equal chords being equidistant is the second favourite. Practice both in standard Given-To Prove-Proof format.
How do I remember whether equal chords are equidistant or equidistant chords are equal?+
Both directions are true (Theorem 10.2 and its converse). Mnemonic: 'Equal length = Equal distance'. If two chords have the same length, their perpendicular distances from centre are equal; if two chords are the same distance from centre, their lengths are equal. Symmetry works both ways in circles.
What is the quickest way to find the length of a chord if I know the radius and the perpendicular distance from the centre?+
Use the formula derived from Pythagoras theorem: Chord length = 2√(r² − d²), where r = radius and d = perpendicular distance from centre. Remember to multiply by 2 because the perpendicular bisects the chord, so you calculate half-length first then double it.
In a cyclic quadrilateral, if I know one angle, how do I find the opposite angle?+
Use the property that opposite angles in a cyclic quadrilateral sum to 180°. So if ∠A is given, then ∠C = 180° − ∠A. Similarly ∠B + ∠D = 180°. This is Theorem 10.11 and is tested in almost every CBSE paper as a direct 1-mark or 2-mark question.
Why is the angle in a semicircle always 90°? How do I prove it?+
A semicircle is an arc that subtends 180° at the centre (a straight line, the diameter). By Theorem 10.8, the angle subtended by an arc at the centre is double the angle at any point on the circle. So angle at the circle = 180°/2 = 90°. This is a direct corollary and a favourite 2-mark question.
Can a quadrilateral with one pair of opposite angles summing to 180° always be cyclic?+
No, you need both pairs of opposite angles to sum to 180°. If only ∠A + ∠C = 180° but ∠B + ∠D ≠ 180°, the quadrilateral cannot be inscribed in a circle. The converse of Theorem 10.11 requires the full condition: opposite angles summing to 180° ⇒ cyclic.
How should I draw the diagram for a circle-geometry proof to get full marks?+
Use a compass for the circle (not freehand), a ruler for chords and radii, label all points with capital letters matching the question, mark right angles with a small square, and use tic marks to show equal lengths (radii). A neat, accurate diagram earns you 0.5-1 mark and prevents logical errors.
What is the difference between 'perpendicular from centre bisects chord' and 'line from centre bisects chord is perpendicular'?+
The first is Theorem 10.3: if a line from centre is perpendicular to a chord, then it bisects the chord. The second is the converse: if a line from centre bisects a chord, it must be perpendicular to the chord. Both are true, but you must state which direction you are using in your proof.
Is Chapter 5 circle geometry important for Class 10 board exams as well?+
Yes, absolutely. Class 10 Chapter 10 'Circles' builds directly on Class 9 concepts, adding tangents, secants, and the tangent-radius perpendicularity theorem. Mastering Class 9 circle theorems now saves you significant effort in Class 10 and also supports coordinate geometry of circles in Class 11.
How can CBSETUTOR.ai help if I am stuck on a circle-geometry problem at 10 p.m. the night before my exam?+
Simply snap a photo of the problem (even a hand-drawn diagram works) and upload it to CBSETUTOR.ai. Within 60 seconds you get a step-by-step solution with theorem names, diagram, and explanation in CBSE language. No waiting for a tutor to reply. Available 24×7, ₹999/month for unlimited doubts across all subjects and classes 6-12. Try the 3-day free trial risk-free.

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