This MCQ module is based on: Playing with Constructions Exercises
TOPIC 32 OF 41
Playing with Constructions Exercises
🎓 Class 6
Mathematics
CBSE
Theory
Ch 8 — Playing with Constructions
⏱ ~35 min
🌐 Language: [gtranslate]
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This mathematics assessment will be based on: Playing with Constructions Exercises
Targeting Class 6 level in Geometry, with Basic difficulty.
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Chapter 8 — Exercises & Summary
Practice each construction with compass and ruler. Show the arcs you drew. Click "Show Answer" to compare your figure with the expected diagram.
Section 8.5 — Rectangle Exercises
Q1. Construct a rectangle in which one of the diagonals divides the opposite angles into 50° and 40°.
Draw side AB of any length. At A, draw ray making 50° with AB (this will be the diagonal). Also draw perpendicular at A (side AD). Diagonal meets perpendicular at C; the 40° angle at C is complementary. Complete the rectangle.
Q2. Construct a rectangle in which one of the diagonals divides the opposite angles into 45° and 45°. What do you observe about the sides?
All four sides are equal — the rectangle is a square.
Q3. Construct a rectangle one of whose sides is 4 cm and the diagonal is of length 8 cm.
Let PQ = 4 cm. Erect perpendicular at Q. From P, draw an arc of radius 8 cm — it cuts the perpendicular at R. Complete rectangle PQRS.
Note: Other side = \(\sqrt{8^2-4^2}=\sqrt{48}=4\sqrt{3}\approx6.93\) cm.
Note: Other side = \(\sqrt{8^2-4^2}=\sqrt{48}=4\sqrt{3}\approx6.93\) cm.
Q4. Construct a rectangle one of whose sides is 3 cm and the diagonal is of length 7 cm.
Same method. AB = 3 cm, diagonal = 7 cm. Other side = \(\sqrt{49-9}=\sqrt{40}\approx 6.32\) cm.
Section 8.6 — House & Related Constructions
Q1. Construct a bigger house in which all sides are of length 7 cm.
Same structure as the 5 cm house but scaled up. Rectangle BCED has sides 7 cm; triangle ABC above has all sides 7 cm (equilateral roof). Apex A is 7 cm from both B and C — use compass arcs of radius 7 cm from B and C.
Q2. Try to recreate 'A Person', 'Wavy Wave' and 'Eyes' using the ideas involved in the 'House' construction.
Each figure combines simple primitives:
• A Person = circle (head, radius 2 cm) + square (body, side 4 cm) + joining line.
• Wavy Wave = two half-circles of equal radius on opposite sides of a line.
• Eyes = two lens shapes made by two arcs meeting at two endpoints, each containing a dark circle (pupil).
• A Person = circle (head, radius 2 cm) + square (body, side 4 cm) + joining line.
• Wavy Wave = two half-circles of equal radius on opposite sides of a line.
• Eyes = two lens shapes made by two arcs meeting at two endpoints, each containing a dark circle (pupil).
Q3. Is there a 4-sided figure in which all sides are equal in length but is not a square? If such a figure exists, construct it.
Yes — a rhombus. All sides equal but angles are not 90°. To construct: draw any side AB. At A, choose an angle (say 60°). Mark D on this ray with AD = AB. From D draw arc of radius = AB; from B draw arc of same radius; they meet at C. Join C to D and B.
Activity: Compass-Only Regular Hexagon
L4 AnalyseMaterials: Compass, ruler, blank paper.
Predict: Using a compass set to a fixed radius, can you mark exactly 6 equally-spaced points around a circle?
- Draw a circle of radius 4 cm with centre O.
- Without changing the compass width, place the tip on any point P on the circle and mark an arc cutting the circle at Q.
- Move the tip to Q and mark the next point R. Continue until you return to P.
- Join the 6 points in order. What shape appears?
A regular hexagon! This happens because the chord length equal to the radius subtends exactly 60° at the centre, and 6 × 60° = 360°.
Competency-Based Questions
Scenario: A classroom is preparing a display where each student must construct one geometric figure: a square, a rectangle, a rhombus, and a house. Only a ruler and compass may be used (no protractor for two of the four figures).
Q1. List which of the four figures can be built using only a compass and ruler (no protractor).
L3 ApplyAll four can be constructed using only a compass and ruler: right angles can be made via perpendicular-bisector arcs. (A protractor is a convenience, not a necessity.)
Q2. Analyse how a rhombus differs from a square even though both have four equal sides.
L4 AnalyseA square has four 90° angles and two equal diagonals. A rhombus has four equal sides but its angles need not be 90°; its diagonals are unequal but still bisect each other at right angles.
Q3. A student says the diagonal of the 3 cm × 4 cm rectangle is 7 cm (just adding sides). Evaluate and give the correct length.
L5 EvaluateWrong. By the Pythagorean theorem the diagonal = \(\sqrt{3^2+4^2}=\sqrt{25}=5\) cm, not 7 cm.
Q4. Design a "big house" pattern: base square 10 cm × 10 cm, equilateral triangle roof of side 10 cm, a centred 3 cm × 4 cm door at the bottom. Describe construction steps.
L6 Create1. Draw base square BCED of side 10 cm. 2. From B and C, draw arcs of 10 cm — they intersect at A; join AB, AC to form the equilateral roof. 3. On DE (bottom side), find the midpoint M. Mark points 1.5 cm left and right of M — these are door corners on DE. Draw perpendiculars of 4 cm up from these — join their tops to form the door. Verify door is 3 cm × 4 cm, centred.
Assertion–Reason Questions
A: Every rhombus is a square.
R: A rhombus has all four sides equal.
R: A rhombus has all four sides equal.
(d) — A is false (a square also needs right angles). R is true.
A: In the house figure with all sides 5 cm, the apex A is 5 cm from both top corners.
R: Two intersecting arcs of radius 5 cm from the top corners locate A.
R: Two intersecting arcs of radius 5 cm from the top corners locate A.
(a) — Both true; R is exactly the construction that satisfies A.
A: A rectangle whose diagonal is twice one of its sides has the other side equal to \(\sqrt{3}\) times the first.
R: By the Pythagorean theorem, \((2s)^2 = s^2 + (\text{other})^2\).
R: By the Pythagorean theorem, \((2s)^2 = s^2 + (\text{other})^2\).
(a) — Both true. From \(4s^2 = s^2 + x^2\) we get \(x^2=3s^2\), so \(x=s\sqrt{3}\).
Chapter 8 — Summary
- A circle is the set of all points at the same distance (radius) from a fixed centre.
- A compass draws circles and arcs of any chosen radius; a ruler draws straight lines.
- A square has four equal sides and four right angles; a rectangle has equal opposite sides and four right angles.
- A rectangle can be constructed given length & breadth, or given a side and a diagonal.
- When a diagonal of a rectangle divides an opposite angle into two equal 45° parts, the rectangle is a square.
- The set of points equidistant from two fixed points A and B forms the perpendicular bisector of AB.
- A rhombus is a 4-sided figure with all sides equal (not necessarily right-angled).
- A rough diagram, arcs, and careful use of compass settings can build complex figures like the "house".
Key Terms
Circle • Centre • Radius • Chord • Square • Rectangle • Rhombus • Diagonal • Perpendicular bisector • Equidistant • Construction • Compass • Ruler.
Frequently Asked Questions
What is the summary of Chapter 8 Playing with Constructions?
Chapter 8 builds geometric drawing skills using ruler and compass: drawing segments, circles, perpendiculars, squares, rectangles, and exploring diagonal properties and equidistant points.
How do you check if a rectangle is drawn correctly?
Measure both pairs of opposite sides for equality, check that each corner is a 90-degree angle, and verify that both diagonals have exactly the same length.
What is the common mistake students make in constructions?
The compass radius shifting while drawing arcs, moving the ruler while drawing a line, or not pressing the needle firmly at the centre. These small errors shift angles and side lengths.
How do construction skills help in later classes?
Class 7 and 8 extend these skills to triangles, angle bisectors, quadrilaterals and Class 9 introduces formal construction proofs and circle constructions.
Should exercises be done on plain or graph paper?
NCERT recommends plain paper so students use the compass and ruler rather than relying on grid lines. This builds real geometric skill.
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