Class 9 Mathematics Chapter 8: Playing with Constructions – Previous Year Questions with Solutions (2020–2025)

When you read Chapter 8 theory once more, you're passive. Your brain registers steps—'draw a line, mark points, join them'—but you don't internalise *why* a compass must stay at a fixed width or *what happens* if you alter angle bisector construction slightly. Past papers force you to do the reverse: you see an exam-style demand ('construct a triangle ABC where AB = 5 cm, BC = 6 cm, AC = 7 cm') and your mind immediately recalls which theorem applies, how many arcs to draw, and where to check accuracy. Moreover, CBSE examiners repeat construction patterns. A perpendicular bisector question asked in 2021 reappears—subtly restyled—in 2024. A triangle construction with an obtuse angle or a specific angle bisector appears across multiple years. By solving 5–7 past papers, you spot these repeats and gain confidence because you've already solved a near-identical problem. Lastly, past papers show you marking expectations: examiners award marks not just for the final drawing but for showing construction lines, labelling, and justification. Theory books don't emphasise this; papers do.

1-mark questions in this chapter test quick recall and basic definitions. They often ask for steps, tools required, or properties of constructions. Expect them to appear in multiple-choice or short-answer format. **Q1: What is the first step in constructing the perpendicular bisector of a line segment AB?** *Answer:* With centre A and radius more than half of AB, draw two arcs—one above and one below AB. (Some papers ask this as 'why do you open the compass more than half the length?' The answer emphasises that the two arcs from both A and B must intersect.) **Q2: To construct an angle of 60°, which angle do you bisect?** *Answer:* An angle of 120° (or construct a 60° angle directly using a compass by marking equal radii). **Q3: State the condition for constructing a triangle given three side lengths.** *Answer:* The sum of any two sides must be greater than the third side (triangle inequality). **Q4: If you are asked to construct triangle ABC with AB = 6 cm, ∠A = 45°, and ∠B = 60°, which angles will you use?** *Answer:* ∠A = 45°, ∠B = 60°, and ∠C = 75° (since angles sum to 180°). This tests whether students calculate the third angle before starting construction. **Q5: What is the difference between constructing an angle bisector and a perpendicular bisector?** *Answer:* An angle bisector divides an angle into two equal parts; a perpendicular bisector divides a line segment into two equal parts *and* is perpendicular (90°) to it.

3-mark questions demand step-by-step construction descriptions or a combination of construction with calculation. Examiners expect you to list steps, show arcs/lines, and often justify why the construction works. **Q1: Construct the perpendicular bisector of a line segment PQ = 7 cm.** *Solution:* (i) Draw PQ = 7 cm. (ii) With centre P and radius > 3.5 cm, draw arcs above and below PQ. (iii) With the same radius and centre Q, draw arcs intersecting the first arcs at points M and N. (iv) Join MN; this is the perpendicular bisector. *Why:* Every point on MN is equidistant from P and Q. **Q2: Construct angle ABC = 75° without using a protractor.** *Solution:* (i) Draw a ray BA. (ii) Construct ∠ = 90° at B (using perpendicular bisector method or 45° + 45°). (iii) Construct ∠ = 60° at B. (iv) The difference gives 30°. Mark 30° from the 90° ray to get 60°. Add 15° (bisect 30°) to get 75°. *Mark allocation:* 1 mark for correct angle, 1.5 marks for steps, 0.5 mark for labelling. **Q3: Construct triangle XYZ where XY = 4 cm, YZ = 5 cm, XZ = 6 cm. Verify using the triangle inequality.** *Solution:* (i) Draw XY = 4 cm. (ii) With centre X and radius 6 cm, draw an arc. (iii) With centre Y and radius 5 cm, draw an arc intersecting the first arc at Z. (iv) Join XZ and YZ. Verify: 4 + 5 = 9 > 6 ✓, 5 + 6 = 11 > 4 ✓, 6 + 4 = 10 > 5 ✓. **Q4: Construct the angle bisector of ∠PQR = 80°.** *Solution:* (i) With centre Q and any radius, cut QP at A and QR at B. (ii) With centres A and B and radius > half of AB, draw arcs intersecting at C. (iii) Join QC; this bisects ∠PQR into two 40° angles. *Justification:* By SSS congruency, triangles QAC and QBC are congruent. **Q5: Construct triangle ABC where AB = 5.5 cm, ∠A = 50°, ∠B = 70°.** *Solution:* (i) Draw AB = 5.5 cm. (ii) At A, construct ∠ = 50°. (iii) At B, construct ∠ = 70°. (iv) Extend these rays to meet at C. Triangle ABC is complete. *Note:* ∠C = 180° − 50° − 70° = 60°.

5-mark questions combine construction with justification or require two constructions in one problem. These are time-intensive but high-reward if practised. **Q1: Construct a triangle ABC where BC = 6.5 cm, ∠B = 60°, ∠C = 75°. Also, construct the perpendicular bisector of BC.** *Full Solution:* (i) Draw BC = 6.5 cm as the base. (ii) At B, construct ∠ = 60° using compass (30° + 30° or bisecting 120°). Draw ray BA along this angle. (iii) At C, construct ∠ = 75° using compass (60° + 15° = bisect 30° and add to 45°). Draw ray CA along this angle. (iv) Extend BA and CA to meet at A. Triangle ABC is complete. ∠A = 180° − 60° − 75° = 45°. (v) Now, to construct perpendicular bisector of BC: With centre B and radius > 3.25 cm, draw arcs above and below BC. Repeat with centre C and the same radius. Join the intersection points; this is the perpendicular bisector. *Marking:* 2 marks for triangle construction, 1.5 marks for angle constructions with proper steps, 1.5 marks for perpendicular bisector. **Q2: Construct triangle PQR such that PQ = 6 cm, QR = 7 cm, PR = 8 cm. Then construct the angle bisector of ∠Q.** *Full Solution:* (i) Draw PQ = 6 cm. (ii) With centre P and radius 8 cm, draw an arc. (iii) With centre Q and radius 7 cm, draw an arc intersecting the first arc at R. (iv) Join PR and QR. Triangle PQR is complete. (v) To bisect ∠Q: With centre Q and any radius, cut QP at M and QR at N. (vi) With centres M and N and equal radius > half MN, draw arcs meeting at L. (vii) Join QL; this is the angle bisector of ∠Q. *Why:* Triangles QML and QNL are congruent (SSS), so ∠MQL = ∠NQL. *Marking:* 2 marks for triangle, 2 marks for angle bisector steps, 1 mark for justification. **Q3: Construct a right-angled triangle ABC with ∠B = 90°, BC = 5 cm, AC = 8 cm (AC is the hypotenuse). Verify using Pythagoras' theorem.** *Full Solution:* (i) Draw BC = 5 cm. (ii) At B, construct a perpendicular (90° angle) using compass or by constructing the perpendicular bisector of a line and rotating it. Draw ray BA perpendicular to BC. (iii) With centre C and radius 8 cm (the hypotenuse), draw an arc intersecting BA at A. (iv) Join AC. Triangle ABC is complete with ∠B = 90°. (v) Measure AB (should be approximately 6.24 cm). (vi) Verify: AB² + BC² = AC² → 6.24² + 5² ≈ 39 + 25 = 64 ≈ 8² ✓ *Marking:* 2 marks for construction, 1.5 marks for perpendicular at B, 1 mark for AC = 8 cm (hypotenuse), 0.5 mark for Pythagoras verification.

The rationalized 2024–25 CBSE Class 9 Mathematics syllabus retains Chapter 8 (Playing with Constructions) but with subtle emphasis changes. Here's what to expect in 2026–27: 1. **More emphasis on digital tools:** While compass-and-straightedge constructions remain the core, examiners now occasionally ask 'How would you verify this construction using geometry software?' This doesn't replace hand-drawing but tests understanding. Expect a 1-mark question like 'Name one geometric property you can check digitally after constructing a perpendicular bisector.' 2. **Justification over rote steps:** Newer papers ask fewer 'list the steps' questions and more 'why does this construction work?' Prepare reasoning based on congruency (SSS, SAS, ASA) and properties of isosceles triangles (used in perpendicular bisector and angle bisector constructions). 3. **Hybrid construction + algebra:** Expect a 3-mark question combining construction with calculation—e.g., 'Construct triangle ABC with AB = 5 cm, ∠A = 45°, ∠B = 60°. Then calculate ∠C and explain why a triangle with ∠A + ∠B + ∠C must equal 180°.' 4. **Fewer 5-mark pure-construction questions:** The new pattern leans toward shorter, concept-heavy questions. However, one or two 5-mark questions per paper still appear, often combining two constructions (e.g., triangle + perpendicular bisector) or asking for construction followed by measurement and verification. 5. **Real-world context:** Look for questions framed as 'A surveyor needs to mark equal distances from two corner pillars...' This tests your ability to apply perpendicular bisector concepts practically, not just mechanically. **Study tip:** Don't just memorise steps. Internalise *why* each arc and line serves a purpose. This shift rewards conceptual clarity over procedural memorisation.

Walking into an exam, you have 15–20 minutes for Chapter 8 questions (if they form part of a mixed-geometry section). Here's a strategic approach: **For 1-mark questions (30 seconds each):** • Read carefully—many 1-mark traps ask 'which angle do you bisect to get 60°?' Students say '60°' but the correct answer is '120°.' • If it's a definition question ('What is a perpendicular bisector?'), write one sentence: 'A line that divides a segment into two equal parts and is perpendicular to it.' **For 3-mark questions (4–5 minutes each):** • Before drawing, note all given data. Example: 'AB = 6 cm, ∠A = 50°, ∠B = 65°' → immediately calculate ∠C = 65°. • Draw lightly in pencil first. If an arc is misplaced, erase and redraw without losing marks. • Label every point (A, B, C, arcs, intersections). Examiners allocate 0.5–1 mark for proper labelling. • List steps in order (not narrative; bullet points are fine). Example: ○ Draw AB = 6 cm. ○ At A, construct ∠ = 50°. ○ Draw ray AC. ○ (And so on.) **For 5-mark questions (7–8 minutes each):** • Identify if it's a 'single construction + verification' (easier) or 'two constructions' (harder). Manage time accordingly. • If combining two constructions (e.g., triangle + angle bisector), complete the first fully, then move to the second. Don't jump between them. • For verification (e.g., Pythagoras), do it *after* construction so measurements are fresh. Write the formula, substitute values, and confirm. **General tips:** • Compass accuracy matters. If the exam allows, sharpen your pencil beforehand. A blunt pencil produces wide arcs that don't intersect cleanly. • If a compass is provided in the exam, test it on scrap paper first to ensure the needle doesn't slip. • For perpendicular bisector and angle bisector questions, always use the same radius twice (step ii and iii). Examiners specifically check this. • If you run out of time, sketch the construction roughly and describe the remaining steps in words. You'll earn partial credit. Start a 3-day free trial at cbsetutor.ai to access video explanations of each construction—watch an expert demo the exact technique, then practise on our interactive geometry canvas.

1. **Can you draw a perpendicular bisector without a protractor?** Practise once more—this is the single most-tested construction. 2. **Can you list the triangle inequality conditions from memory?** (Sum of any two sides > third side.) 3. **Do you know how to construct 30°, 45°, 60°, 75°, 90°, 120° using compass and straightedge?** Test yourself. 4. **Have you solved at least 3 full past papers (not just this chapter)?** Timing and accuracy improve with repetition. 5. **Do you understand *why* a perpendicular bisector works?** (Every point on it is equidistant from the endpoints—use isosceles triangle property.) 6. **Can you construct a triangle given (a) three sides (SSS), (b) two sides and included angle (SAS), (c) one side and two angles (ASA)?** All three types appear across papers. 7. **Do you check your construction?** For example, after drawing the angle bisector, measure both resulting angles with a protractor to confirm they're equal (even if you lose the protractor before submission, this practice ensures accuracy). 8. **Have you written down the marking criteria for each question type?** (1-mark = definition; 3-mark = steps + labelling; 5-mark = construction + justification or verification.) If you've ticked all eight, you're exam-ready. Good luck!