Class 9 Mathematics Chapter 9 Constructions: Solved Previous Year Questions (2020–2025)

Many Class 9 students spend hours drawing neat construction diagrams in their notebooks but still lose marks in exams. Why? Because textbook reading teaches you *what* a construction is, but past papers teach you *how examiners expect it done*. When you solve a PYQ on constructing a triangle by SSS, you discover that marks are awarded step-by-step: (1) drawing the first side to scale, (2) opening compass for two arcs, (3) identifying the intersection point, (4) labeling angles and sides. A five-minute textbook example won't show you all these breakpoints. Past papers also expose the language examiners use. If a question says 'construct a triangle ABC with AB = 5 cm, BC = 6 cm, AC = 7 cm,' you must know instantly which rule applies (SSS). Over 5 years of CBSE papers, certain constructions appear repeatedly: perpendicular bisectors of sides, angle bisectors, parallel lines using alternate angles, and RHS triangles. By solving 10–15 PYQs, you'll recognize these patterns and approach any new construction confidently. You'll also learn time management — construction questions require 3–5 minutes to draw and label neatly, and practicing past papers trains you to work at exam speed without sacrificing accuracy. Finally, mistakes you make on practice PYQs are far less costly than mistakes on the real exam. Work through the papers below, mark your own diagram quality, and refine your technique.

One-mark questions in Constructions typically ask for a single definition, rule, or a very short construction verification. These are knowledge-check questions. Here are five commonly repeated types: **Q1. What is the full form of SSS in triangle construction?** Ans: Side–Side–Side. When all three sides of a triangle are given, you use the SSS rule to construct the triangle uniquely. Example: If AB = 4 cm, BC = 5 cm, CA = 6 cm are given, construct triangle ABC using SSS. **Q2. How many conditions are required to construct a triangle uniquely?** Ans: Minimum three independent measurements. These can be three sides (SSS), two sides and an included angle (SAS), two angles and a side (ASA), or the right angle and two sides in a right triangle (RHS). **Q3. What is the difference between a perpendicular and a perpendicular bisector?** Ans: A perpendicular is a line at 90° to another line. A perpendicular bisector is a line perpendicular to a segment AND passing through its midpoint, dividing it into two equal parts. **Q4. State the axiom used to construct parallel lines using alternate angles.** Ans: If a transversal intersects two lines such that alternate interior angles are equal, then the two lines are parallel (Alternate Angle Axiom). **Q5. How many arcs do you draw to bisect an angle using compass and straightedge?** Ans: Three arcs total — one arc from the vertex intersecting both arms, then two arcs from the intersection points on the arms to meet at the bisector point.

Three-mark Constructions questions ask you to perform a full construction with clear steps. Marks are split: typically 1 mark for correct method/steps, 1 for accurate drawing, 1 for correct labeling and measurements. Here are five high-frequency types: **Q1. Construct a triangle ABC where AB = 5 cm, AC = 6 cm, and ∠BAC = 60°. [3 marks]** Ans: (1) Draw AB = 5 cm using ruler. (2) At A, construct a 60° angle using compass (30°–60°–90° set or protractor). (3) On the 60° line, mark point C such that AC = 6 cm. (4) Join BC. Triangle ABC is ready. Label all sides and angle. *Method mark: drawing AB and angle correctly; Accuracy mark: measurements exact; Label mark: all vertices and angle marked.* **Q2. Construct a right-angled triangle ABC with hypotenuse AC = 7 cm and one side BC = 5 cm. [3 marks]** Ans: This is RHS (Right angle–Hypotenuse–Side). (1) Draw AC = 7 cm (hypotenuse). (2) Find midpoint M of AC. (3) Draw a semicircle with AC as diameter. (4) Mark point B on the semicircle such that BC = 5 cm. (5) Join AB. The angle at B is automatically 90° (angle in semicircle). Label and verify AB² + BC² = AC². **Q3. Draw a line segment AB = 6 cm. Construct the perpendicular bisector of AB. [3 marks]** Ans: (1) Draw AB = 6 cm. (2) Open compass more than half of AB. (3) Draw arcs above and below AB from point A. (4) Keeping same radius, draw arcs from B intersecting the first arcs at points P and Q. (5) Join P and Q. Line PQ is the perpendicular bisector of AB. It cuts AB at midpoint (3 cm from both A and B) at a right angle. **Q4. Construct an angle of 45° at a point P on a line segment PQ without using a protractor. [3 marks]** Ans: (1) At P, construct a perpendicular to PQ using compass (two arcs equidistant from P on PQ, then arcs above to find perpendicular line). (2) Bisect this 90° angle: place compass at P, draw an arc cutting both the perpendicular and PQ, then bisect to get 45°. (3) Label the 45° angle ray as PR. **Q5. Draw a line segment AB = 8 cm. At point A, construct a line parallel to a given line l passing through B, using the angle axiom. [3 marks]** Ans: (1) Draw AB = 8 cm and line l through B not parallel to AB. (2) At B, measure the angle between l and BA. (3) At A, on the opposite side of AB, construct an equal angle (alternate angle). (4) The line through A making this angle is parallel to l. Verify with a ruler or set square.

Five-mark questions demand a complete, multi-step construction with proof or verification. These are usually combined constructions: building a triangle and then constructing something within or on it (angle bisector, perpendicular, parallel). Here are three detailed solutions: **Q1. Construct a triangle ABC with AB = 6 cm, ∠A = 45°, ∠B = 60°. Also construct the perpendicular bisector of side AC. [5 marks]** Solution: (i) Draw AB = 6 cm using a ruler. (ii) At A, construct 45° using compass: draw an arc intersecting AB and another line through A; bisect the right angle (90°) and then bisect one resulting 45° to verify, or use a 45°–45°–90° set. (iii) At B, construct 60° using a compass: draw an arc, and mark two intersections 60° apart (equilateral triangle method or protractor). (iv) Extend both angle rays until they meet at point C. (v) Triangle ABC is now constructed. Verify: ∠C = 180° − 45° − 60° = 75°. (vi) To construct the perpendicular bisector of AC: find the midpoint by constructing a perpendicular bisector using compass (arcs from A and C, equal radius > AC/2, intersecting above and below). (vii) The bisector line crosses AC at its midpoint and is perpendicular (90°). Mark and label this clearly. Marks: (1) AB drawn correctly, (2) angle at A correct, (3) angle at B correct, (4) accurate intersection at C, (5) perpendicular bisector method and accuracy. **Q2. Construct a triangle PQR where PQ = 7 cm, QR = 6 cm, and RP = 5.5 cm (SSS). Then construct the angle bisector of angle Q. [5 marks]** Solution: (i) Draw PQ = 7 cm. (ii) Open compass to 5.5 cm. From P, draw an arc above PQ. (iii) Open compass to 6 cm. From Q, draw an arc intersecting the first arc at R. (iv) Join PR and QR. Triangle PQR is complete. Measure and label all three sides. (v) To bisect angle Q: place compass point at Q, draw an arc intersecting QP and QR at two points X and Y. (vi) Open compass to more than half of XY. From X and Y, draw intersecting arcs inside the angle at point Z. (vii) Join QZ. This is the angle bisector of ∠Q. Mark the equal angles on both sides (optional: verify they are equal using a protractor). Marks: (1) all three sides drawn to exact length, (2) triangle closure accurate, (3) correct identification of SSS rule, (4) angle bisector method precise, (5) labeling and final diagram clarity. **Q3. Construct a rectangle ABCD with length AB = 8 cm and width BC = 5 cm. Also construct the perpendicular from vertex D to the diagonal AC. [5 marks]** Solution: (i) Draw AB = 8 cm using a ruler. (ii) At B, construct a 90° angle using compass: place compass at B, draw equal arcs on AB and a perpendicular line, then mark BC = 5 cm on the perpendicular. (iii) At A, construct another 90° and mark AD = 5 cm. (iv) At D, draw a 90° and mark DC = 8 cm. The rectangle ABCD is complete. (v) Draw diagonal AC by joining A and C. (vi) To drop a perpendicular from D to AC: place compass at D, draw an arc intersecting AC at two points E and F. (vii) From E and F, draw intersecting arcs below AC (on the side of D) meeting at point G. (viii) Join DG. DG is perpendicular to AC. Mark the 90° angle at the intersection point. Marks: (1) AB and angles at A and B accurate, (2) sides BC and AD correct, (3) DC accurate, (4) diagonal AC drawn clearly, (5) perpendicular from D to AC method and accuracy. All 5-mark constructions require: (a) precise measurements, (b) correct geometric method, (c) clear labeling of all points and angles, (d) where applicable, proof or verification. Begin practicing these patterns at cbsetutor.ai, which offers step-by-step construction walkthroughs with real-time feedback on diagram accuracy.

The 2024–25 CBSE rationalization reduced the breadth of geometry but increased depth and reasoning. For Constructions in Class 9, here are the key shifts to watch: **Shift 1: More emphasis on 'Why' alongside 'How'** Older papers often asked: 'Construct a triangle ABC with…' New papers increasingly ask: 'Can a triangle be constructed with the given measurements? Justify using a construction rule.' This means you must know not just *how* to construct, but *why* a particular set of measurements works (or doesn't). Example: 'Can you construct a triangle with sides 3 cm, 4 cm, 8 cm? Why or why not?' Answer: No, because 3 + 4 = 7 < 8, violating the triangle inequality theorem. **Shift 2: Increased use of combined constructions** Older papers: 'Construct a triangle ABC with AB = 5 cm, BC = 6 cm, CA = 7 cm.' New papers: 'Construct triangle ABC as above. Then construct the perpendicular from C to AB, meeting AB at point D. Measure CD.' This tests multiple skills in one question and aligns with the NCERT emphasis on integrated problem-solving. **Shift 3: Stronger focus on parallel lines and angle-chasing** With the removal of some algebra-heavy chapters, parallel line constructions and their angle properties are now tested more rigorously. Expect: 'Construct a line parallel to AB through an external point P, and verify using alternate angles or co-interior angles.' **Shift 4: Real-world application language** New CBSE papers use language like: 'A surveyor needs to construct a triangle representing a plot of land with sides 12 m, 15 m, 18 m. Construct the triangle on a scale of 1 cm = 2 m.' This contextualizes constructions and requires you to scale down measurements before drawing. **Shift 5: Likely reduction in 5-mark pure construction, increase in 3-mark multi-step** Feedback from early 2025 papers suggests CBSE may move away from standalone 5-mark construction questions. Instead, 3-mark 'construct and measure' or 'construct and verify' questions are more frequent. However, 5-mark questions combining construction with angle/length calculations remain common. **Preparation tip:** Practice 'proof-based' construction: after every construction, write one sentence justifying why your method works using a theorem or axiom. This aligns with the new pattern.

Constructions are unusual in Class 9 exams because they test accuracy, not just conceptual knowledge. A small error in a compass opening or an inaccurate angle can lose you marks. Here's a battle-tested strategy to maximize your score in 25–30 minutes (typical time for all Constructions questions in a full paper): **Step 1: Read the entire construction once. (1 minute)** Identify: (a) What shape am I constructing? Triangle, angle, parallel, perpendicular? (b) What are the given measurements? (c) What additional construction is required (bisectors, perpendiculars, etc.)? Write a quick note: 'Triangle ABC, SSS rule, sides 5–6–7 cm.' **Step 2: Identify the rule. (1 minute)** Is this SSS, SAS, ASA, RHS, or a composite (triangle + perpendicular)? Look for keywords: 'all three sides' → SSS, 'two sides and included angle' → SAS, 'two angles and a side' → ASA, 'right angle and hypotenuse and one side' → RHS. If unclear, re-read the given data and match to a rule. **Step 3: Plan the sequence before drawing. (2 minutes)** Write the steps on a rough paper first: (1) Draw AB = 5 cm. (2) From A, draw arc 6 cm radius. (3) From B, draw arc 7 cm radius. (4) Mark intersection as C. (5) Join AC and BC. (6) Label all sides and angles. This prevents mid-diagram mistakes. **Step 4: Start with the primary construction. (12–15 minutes for a full triangle)** Draw the first side using a ruler at the top-left of your answer space, leaving room for the diagram to grow. Use a sharp pencil (HB or B). Make all arcs light and thin so they're visible but not messy. Mark intersection points clearly with small dots. Once you've drawn all three sides, erase the arc lines lightly with a clean eraser (arcs are construction aids, not part of the final answer in some exam formats — check your specific paper's instruction). **Step 5: Label all points and measurements. (3 minutes)** Label vertices (A, B, C, P, Q, R, etc.) at or very close to the points. Write side lengths next to sides: 'AB = 5 cm'. Write angles if required. Use a ruler to place labels neatly. Misplaced labels lose marks. **Step 6: Perform secondary constructions (if any). (5–8 minutes)** If the question asks for a perpendicular bisector, angle bisector, or parallel line, follow the same step-by-step method. Use compass arcs sparingly and intentionally. Again, label clearly. **Step 7: Verify measurements (final 2 minutes)** If time permits, use a ruler to check one or two side lengths. If a measurement is off by more than 2 mm, you have a problem. If you catch this late, don't panic — state in your answer 'Due to drawing limitations, slight variations may occur' or simply leave it; examiners account for small drawing errors. **Time buffer:** If your construction finishes in 20 minutes for a 5-mark question, use the remaining 5 minutes to (a) darken the final diagram lines so they're very clear, (b) double-check labels, (c) if composite construction, verify that secondary parts are accurate. Do not rush; neatness and accuracy are 50% of the marks. **Common mistakes to avoid:** (1) Opening compass inaccurately — re-check against the ruler before each arc. (2) Forgetting to label — examiners can't award marks if they don't know what each line represents. (3) Drawing construction arcs too dark — they clutter the diagram. (4) Measuring angles with a protractor when you should use compass bisection (unless explicitly asked). (5) Not verifying the rule before starting — re-read the given data once more before you pick up the compass. With consistent practice on past papers, constructions shift from your weakest topic to your highest-scoring one. Aim for 95%+ accuracy by the time you sit the real exam.

Constructions (Chapter 9 in the 2024–25 rationalized NCERT Class 9) covers fundamental geometric drawing skills using compass, straightedge (unmarked ruler), and sometimes a protractor. The chapter is divided into four core areas, each tested regularly in exams: **1. Construction of Triangles** Given three independent pieces of information, you construct a unique triangle. The four conditions are: (a) SSS (Side–Side–Side): all three sides known, (b) SAS (Side–Angle–Side): two sides and the included angle known, (c) ASA (Angle–Side–Angle): two angles and the included side known, (d) RHS (Right angle–Hypotenuse–Side): a right triangle with hypotenuse and one other side. Each condition guarantees a unique triangle (up to congruence). Exams test your ability to identify which condition applies and execute the construction with precision. **2. Construction of Perpendiculars** A perpendicular is a line at 90° to another line or segment. Three sub-types are tested: (a) Perpendicular to a line at a given point on the line (using 90° angle construction), (b) Perpendicular from a point outside a line to that line (shortest distance), (c) Perpendicular bisector of a segment (passes through midpoint at 90°). All are constructed using compass arcs based on the equal-radius principle. **3. Construction of Angle Bisectors** An angle bisector divides an angle into two equal parts. Constructed by opening compass from the vertex, drawing arcs on both arms, then drawing equal arcs from these points to meet inside the angle. The line from the vertex through this intersection point is the bisector. Often combined with triangle constructions: 'Construct triangle ABC and then bisect angle A.' **4. Construction of Parallel Lines** Two lines are parallel if they never meet. Constructed using the Alternate Angle Axiom: if a transversal (line cutting two lines) makes equal alternate angles, the two lines are parallel. Alternative method: use corresponding angles or co-interior angles (supplementary). This is tested with prompts like: 'Construct a line parallel to AB through point P' and 'Verify using angles.' All constructions rely on compass precision and ruler straightness. The NCERT emphasizes not just the final diagram but the step-by-step method and the geometric principle behind each step. Marks are awarded for: (1) correct method (stating the rule used), (2) accurate execution (measurements within 1–2 mm), (3) clear labeling. In the new 2026–27 syllabus, expect questions that integrate multiple constructions and ask for reasoning or verification.