Class 6 : Maths ( English ) – Lesson 8. Playing with Constructions
EXPLANATION AND ANALYSIS
🌿 Explanation & Analysis
🔵 1. Introduction to geometrical constructions
In geometry, constructions mean drawing figures accurately by following fixed rules using specific instruments. These drawings are not freehand sketches; instead, they are precise and based on logic. In this chapter, students learn how to draw lines, angles, and circles carefully so that their lengths and angles are exact.
🧠 Constructions help us see geometry instead of only imagining it. They turn abstract ideas into clear visual forms.
✏️ Note: In school geometry, constructions are done using only a ruler (straightedge) and a compass.
💡 Concept: A geometrical construction is an exact drawing made by logical steps, not by estimation.
🔵 2. Why do we need constructions?
Geometrical constructions play an important role in learning geometry because they help students:
🟢 Understand shapes and angles clearly
🟢 Visualise relationships between lines and points
🟢 Develop accuracy, patience, and logical thinking
🟢 Prepare for advanced topics like triangles and circles
➡️ Geometry becomes meaningful when students construct figures themselves instead of memorising properties.
🔵 3. Tools used in geometrical constructions
🟢 Ruler (Straightedge)
A ruler is used to draw straight lines and line segments 📏. It is not used for measuring during constructions; its purpose is only to draw straight paths.
🟡 Compass
A compass is used to draw circles and arcs and to copy lengths accurately 🧭. It ensures that distances remain exactly the same.
🔴 Pencil
A sharp pencil is important for neat and thin lines ✏️. Thick or dark lines can hide intersection points.
✏️ Note: While copying a length or angle, the opening of the compass should not be changed.
🔵 4. Constructing a line segment of a given length
To construct a line segment of a given length, we follow these logical steps:
🔵 Draw a straight line using a ruler
🔵 Mark a starting point
🔵 Use a compass to mark the required length from that point
🧠 This method ensures that the drawn line segment is exactly equal to the given length.
💡 Concept: Copying a length means transferring the same distance using a compass without measuring.
🔵 5. Constructing a circle of a given radius
A circle is a closed curve in which every point is at the same distance from a fixed point called the centre.
Conceptual steps:
🔵 Mark the centre point
🔵 Open the compass to the given radius
🔵 Rotate the compass completely to draw the circle
✔️ All points on the circle are equally distant from the centre.
✏️ Note: The radius of the circle must remain fixed throughout the construction.
🔵 6. Perpendicular bisector of a line segment
A perpendicular bisector is a line that divides a line segment into two equal parts and makes a right angle (90°) with it.
Key ideas:
🔵 The compass is opened to more than half the length of the segment
🔵 Arcs are drawn from both endpoints
🔵 The points where arcs intersect are joined
🧠 This construction is very useful in symmetry and in making triangles.
💡 Concept: A perpendicular bisector cuts a segment into two equal halves at right angles.
🔵 7. Constructing an angle equal to a given angle
Sometimes we need to copy an angle exactly. This can be done using a compass and ruler.
Basic idea:
🔵 Draw a ray
🔵 Draw an arc from the vertex of the given angle
🔵 Draw the same arc on the new ray
🔵 Join the intersection point
✔️ The angle formed is exactly equal to the given angle.
✏️ Note: Do not change the compass width while copying the arc.
🔵 8. Constructing a perpendicular to a line from a point on it
A perpendicular is a line that makes an angle of 90° with another line.
Concept involved:
🔵 Take a point on the given line
🔵 Draw arcs on both sides of the point
🔵 Draw arcs from these points to intersect
🔵 Join the intersection with the given point
🧠 This construction helps in drawing accurate right angles.
🔵 9. Constructing a perpendicular to a line from a point outside it
When the point does not lie on the line, the construction is slightly different.
Key steps in idea:
🔵 Draw arcs from the given point to cut the line
🔵 From the cut points, draw arcs that intersect
🔵 Join the intersection point with the given point
✔️ The new line drawn is perpendicular to the given line.
🔵 10. Importance of accuracy in constructions
Accuracy is the most important part of geometrical constructions.
🔴 Small mistakes can lead to incorrect results
🔴 Changing compass width can spoil the construction
🔴 Guessing lengths should be avoided
🧠 Careful drawing improves understanding and correctness.
✏️ Note: Neat constructions also help in scoring better marks in exams.
🔵 11. Real-life applications of constructions
Geometrical constructions are widely used in real life:
🏗️ Architecture and building design
🧭 Engineering drawings
🗺️ Map making and layouts
🧠 Planning and designing tools
➡️ These skills are useful beyond the classroom.
🔵 12. Learning geometry by doing
Playing with constructions makes geometry more:
🟢 Visual
🟢 Logical
🟢 Interesting
🟢 Easy to understand
✔️ When students construct figures themselves, geometry becomes enjoyable and meaningful.
💡 Concept: Learning by doing strengthens understanding and confidence.
Summary
Geometrical constructions are exact drawings made using a ruler and a compass. They help students understand geometric ideas clearly and accurately. Through constructions, students learn how to draw line segments, circles, perpendicular bisectors, equal angles, and perpendicular lines by following logical steps.
Accuracy and careful handling of instruments are essential because even small errors can change the entire figure. Constructions form the foundation for advanced geometry topics such as triangles, circles, and symmetry. They are also useful in real-life fields like architecture, engineering, and design. Learning constructions makes geometry practical, visual, and enjoyable.
📝 Quick Recap
🔵 Constructions are accurate geometric drawings
🟢 Ruler and compass are basic tools
🟡 Logical steps must be followed carefully
🔴 Accuracy and neatness are very important
✔️ Constructions build a strong base for geometry
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TEXTBOOK QUESTIONS
🌿 CONSTRUCTION
🔒 ❓ Figure it Out
- What radius should be taken in the compass to get this half circle?
What should be the length of AX?
📌 ✅ Answer:
🔹 AB = 8 cm and the two waves are identical.
🔹 So AX is half of AB.
🔹 AX = 8/2
🔹 AX = 4 cm
🔹 The half-circle is drawn on diameter AX.
🔹 Radius = AX/2
🔹 Radius = 4/2
🔹 Radius = 2 cm
🔹 Final: Radius in compass = 2 cm, and AX = 4 cm.
🔒 ❓
2. Take a central line of a different length and try to draw the wave on it.
📌 ✅ Answer:
🔹 Step 1: Draw a central line AB of any length you like (for example 10 cm, 12 cm, 14 cm, etc.).
🔹 Step 2: To make 2 identical waves, mark the midpoint X of AB.
🔹 Step 3: Now AX = XB, so both halves are equal.
🔹 Step 4: Use AX as the diameter for the first half-circle (draw it above AB).
🔹 Step 5: Use XB as the diameter for the second half-circle (draw it below AB).
🔹 Step 6: Keep the radius for each half-circle = (its diameter)/2.
🔹 Key idea: If the two diameters are equal, the two waves will be identical.
🔒 ❓
3. Try to recreate the figure where the waves are smaller than a half circle (as appearing in the neck of the figure, ‘A Person’).
The challenge here is to get both the waves to be identical.
This may be tricky!
📌 ✅ Answer:
🔹 Meaning: “Smaller than a half circle” here means the curve is flatter (not a perfect semicircle).
🔹 To make both waves identical, we must keep
🔸 the same chord length for each wave
🔸 the same radius for each wave
🔹 Step 1: Draw the central line AB (any length you choose).
🔹 Step 2: Mark midpoint X so that AX = XB.
🔹 Step 3: Decide a radius r for the curve such that r is greater than AX/2.
🔸 This makes the arc flatter than a semicircle.
🔹 Step 4: Make the first arc from A to X (above the line) with radius r:
🔸 Construct the perpendicular bisector of AX.
🔸 On this bisector, mark a point O above AB such that OA = r and OX = r.
🔸 With center O and radius r, draw the arc from A to X.
🔹 Step 5: Make the second identical arc from X to B (below the line) with the same radius r:
🔸 Construct the perpendicular bisector of XB.
🔸 On this bisector, mark a point O’ below AB such that O’X = r and O’B = r.
🔸 With center O’ and radius r, draw the arc from X to B.
🔹 Why this works:
🔸 AX = XB gives equal “base length” (equal chords).
🔸 Same radius r gives equal “curvature”.
🔸 So both waves become perfectly identical.
🌿 SQUARES AND RECTANGLES
🔒 ❓ Q1. Figure it Out
Draw the rectangle and four squares configuration (shown in Fig. 8.3) on a dot paper.
What did you do to recreate this figure so that the four squares are placed symmetrically around the rectangle? Discuss with your classmates.
📌 ✅ Answer:
🔹 First, I drew the given rectangle on the dot paper by joining dots so that opposite sides are equal and parallel.
🔹 I identified the midpoints of each side of the rectangle using the dot grid.
🔹 On each side of the rectangle, I constructed a square outward using that side as the base.
🔹 I kept the side length of all four squares equal to the corresponding side of the rectangle.
🔹 I ensured symmetry by placing one square on each side of the rectangle—top, bottom, left, and right.
🔹 The dot grid helped me keep equal distances and right angles.
🔹 Because all squares have equal sides and are placed uniformly around the rectangle, the whole figure becomes symmetrical.
🔒 ❓ Q2. Figure it Out
Identify if there are any squares in this collection. Use measurements if needed.
📌 ✅ Answer:
🔹 A square has
🔸 all four sides equal
🔸 all four angles equal to 90°
🔹 Figure A:
🔸 All sides appear equal when counted on the dot grid.
🔸 The corners are right angles (the slope changes symmetrically).
🔸 So, A is a square (rotated square).
🔹 Figure B:
🔸 The side lengths are not all equal when counted using dots.
🔸 So, B is not a square.
🔹 Figure C:
🔸 Opposite sides are equal, but adjacent sides are of different lengths.
🔸 So, C is a rectangle, not a square.
🔹 Figure D:
🔸 All sides are not equal and angles are not right angles.
🔸 So, D is neither a square nor a rectangle.
🔹 Final conclusion:
🔸 Only figure A is a square.
🔒 ❓ Think
Is it possible to reason out if the sides are equal or not, and if the angles are right or not without using any measuring instruments in the above figure?
Can we do this by only looking at the position of corners in the dot grid?
📌 ✅ Answer:
🔹 Yes, it is possible to reason without measuring instruments.
🔹 On a dot grid, we can count the number of dot-spaces between corners to compare side lengths.
🔹 If all sides cover the same number of dot-spaces, the sides are equal.
🔹 Right angles can be identified if one side is horizontal and the next is vertical, or if slopes change symmetrically.
🔹 By checking the relative positions of the corners on the grid, we can decide whether a shape is a square or rectangle.
🔹 So, careful observation of dot positions is enough to reason about side lengths and angles.
🔒 ❓ Q3. Figure it Out
Draw at least 3 rotated squares and rectangles on a dot grid. Draw them such that their corners are on the dots.
Verify if the squares and rectangles that you have drawn satisfy their respective properties.
📌 ✅ Answer:
🔹 I first selected four dots that form a rotated shape instead of horizontal or vertical sides.
🔹 For a rotated square:
🔸 I chose four points such that all sides have equal dot-distance.
🔸 I checked that all angles are right angles by observing symmetry.
🔹 For a rotated rectangle:
🔸 I chose four dots such that opposite sides are equal but adjacent sides are unequal.
🔸 I verified that opposite sides are parallel and angles are right angles.
🔹 I repeated this process to draw at least three such shapes.
🔹 Verification results:
🔸 Squares satisfied all properties: equal sides and right angles.
🔸 Rectangles satisfied their properties: opposite sides equal and all angles right angles.
🔹 Thus, the rotated figures also correctly follow their shape properties.
🌿 CONSTRUCTING SQUARES AND RECTANGLES
🔒 ❓ Q1. Construct
Draw a rectangle with sides of length 4 cm and 6 cm. After drawing, check if it satisfies both the rectangle properties.
📌 ✅ Answer:
🔹 Step 1: Draw a line segment AB = 6 cm using a ruler.
🔹 Step 2: At point A, draw a perpendicular line and mark AD = 4 cm.
🔹 Step 3: At point B, draw a perpendicular line on the same side of AB and mark BC = 4 cm.
🔹 Step 4: Join points C and D to complete rectangle ABCD.
🔹 Checking rectangle properties:
🔹 Opposite sides are equal
🔸 AB = CD = 6 cm
🔸 AD = BC = 4 cm
🔹 All angles are right angles
🔸 ∠A = ∠B = ∠C = ∠D = 90°
🔹 Conclusion:
🔹 The drawn figure satisfies both rectangle properties.
🔒 ❓ Q2. Construct
Draw a rectangle of sides 2 cm and 10 cm. After drawing, check if it satisfies both the rectangle properties.
📌 ✅ Answer:
🔹 Step 1: Draw a line segment AB = 10 cm.
🔹 Step 2: At point A, draw a perpendicular and mark AD = 2 cm.
🔹 Step 3: At point B, draw a perpendicular on the same side of AB and mark BC = 2 cm.
🔹 Step 4: Join points C and D to complete rectangle ABCD.
🔹 Checking rectangle properties:
🔹 Opposite sides are equal
🔸 AB = CD = 10 cm
🔸 AD = BC = 2 cm
🔹 All angles are right angles
🔸 Each interior angle measures 90°
🔹 Conclusion:
🔹 The figure drawn is a rectangle and satisfies both rectangle properties.
🔒 ❓ Q3. Try This
Is it possible to construct a 4-sided figure in which
• all the angles are equal to 90° but
• opposite sides are not equal?
📌 ✅ Answer:
🔹 A 4-sided figure with all angles equal to 90° must be a rectangle.
🔹 In a rectangle, opposite sides are always equal by definition.
🔹 If opposite sides are not equal, the figure cannot have all angles equal to 90°.
🔹 Conclusion:
🔹 It is not possible to construct such a figure.
🌿 EXPLORING DIAGONALS OF RECTANGLES AND SQUARES
🔒 ❓ Q1. Construct
Construct a rectangle in which one of the diagonals divides the opposite angles into 50° and 40°.
📌 ✅ Answer:
🔹 Step 1: Draw a line segment AB of any convenient length.
🔹 Step 2: At point A, draw two rays such that the angle between them is 90°.
🔹 Step 3: Inside this right angle at A, draw a ray AC such that
🔸 ∠BAC = 50°
🔸 ∠CAD = 40°
🔹 Step 4: Choose a point C on ray AC.
🔹 Step 5: Through point C, draw a line parallel to AB.
🔹 Step 6: Through point B, draw a line parallel to AD.
🔹 Step 7: Let these two parallel lines meet at point D.
🔹 ABCD is the required rectangle.
🔹 Verification:
🔹 All angles of rectangle are 90°.
🔹 Diagonal AC divides angle A into 50° and 40°.
🔹 Hence, the construction is correct.
🔒 ❓ Q2. Construct
Construct a rectangle in which one of the diagonals divides the opposite angles into 45° and 45°.
What do you observe about the sides?
📌 ✅ Answer:
🔹 Step 1: Draw a line segment AB of any length.
🔹 Step 2: At point A, draw a right angle.
🔹 Step 3: Draw a ray AC inside the right angle such that
🔸 ∠BAC = 45° and ∠CAD = 45°.
🔹 Step 4: Take any point C on AC.
🔹 Step 5: Through C, draw a line parallel to AB.
🔹 Step 6: Through B, draw a line parallel to AD to meet the previous line at D.
🔹 Observation:
🔹 The diagonal divides the right angle into two equal angles.
🔹 This happens only when adjacent sides are equal.
🔹 So, the rectangle formed is actually a square.
🔹 Hence, all sides of the rectangle are equal.
🔒 ❓ Q3. Construct
Construct a rectangle one of whose sides is 4 cm and the diagonal is of length 8 cm.
📌 ✅ Answer:
🔹 Step 1: Draw a line segment AB = 4 cm.
🔹 Step 2: At point A, draw a perpendicular to AB.
🔹 Step 3: On this perpendicular, mark a point D such that AD is less than 8 cm.
🔹 Step 4: With centre B and radius 8 cm, draw an arc to cut the perpendicular at point C.
🔹 Step 5: Through point C, draw a line parallel to AB.
🔹 Step 6: Through point B, draw a line parallel to AD.
🔹 Step 7: Let the two lines meet at point D.
🔹 ABCD is the required rectangle with side 4 cm and diagonal 8 cm.
🔒 ❓ Q4. Construct
Construct a rectangle one of whose sides is 3 cm and the diagonal is of length 7 cm.
📌 ✅ Answer:
🔹 Step 1: Draw a line segment AB = 3 cm.
🔹 Step 2: At point A, draw a perpendicular to AB.
🔹 Step 3: On this perpendicular, mark a ray AD.
🔹 Step 4: With centre B and radius 7 cm, draw an arc to cut AD at point C.
🔹 Step 5: Through point C, draw a line parallel to AB.
🔹 Step 6: Through point B, draw a line parallel to AD.
🔹 Step 7: Let the two parallel lines meet at point D.
🔹 ABCD is the required rectangle.
🌿 POINTS EQUIDISTANT FROM TWO GIVEN POINTS
🔒 ❓ Q1. Construct
Construct a bigger house in which all the sides are of length 7 cm.
📌 ✅ Answer:
🔹 The “house” figure consists of a square base and a triangular roof on top.
🔹 Step 1: Draw a line segment AB = 7 cm.
🔹 Step 2: At point A, draw a perpendicular and mark AD = 7 cm.
🔹 Step 3: Through point D, draw a line parallel to AB.
🔹 Step 4: Through point B, draw a line parallel to AD to meet the previous line at point C.
🔹 ABCD is a square, so all its sides are 7 cm.
🔹 Step 5: Find the midpoint M of the top side DC.
🔹 Step 6: With centre D and radius 7 cm, draw an arc above DC.
🔹 Step 7: With centre C and radius 7 cm, draw another arc to cut the first arc at point E.
🔹 Step 8: Join DE and CE.
🔹 DE = CE = DC = 7 cm
🔹 So, the roof also has sides of length 7 cm.
🔹 Thus, a bigger house is constructed in which all sides are 7 cm.
🔒 ❓ Q2. Construct
Try to recreate ‘A Person’, ‘Wavy Wave’, and ‘Eyes’ from the section ‘Artwork’, using ideas involved in the ‘House’ construction.
📌 ✅ Answer:
🔹 The key idea in the house construction is using equal lengths and arcs with the same radius.
🔹 Recreating ‘A Person’:
🔹 Use straight line segments of equal length for the body, arms, and legs.
🔹 Use the compass with the same opening to draw equal arcs for the head and shoulders.
🔹 Recreating ‘Wavy Wave’:
🔹 Divide the central line into equal parts, just like equal sides in the house.
🔹 Use the same compass radius to draw identical half-circles or arcs alternately above and below the line.
🔹 Recreating ‘Eyes’:
🔹 Use equal line segments for the eye length.
🔹 With the same compass radius, draw intersecting arcs from both ends to form the eye shape.
🔹 In all cases, keeping lengths and radii equal helps in making the figures neat and symmetric.
🔒 ❓ Q3. Construct
Is there a 4-sided figure in which all the sides are equal in length but is not a square?
If such a figure exists, can you construct it?
📌 ✅ Answer:
🔹 Yes, such a figure exists. It is called a rhombus.
🔹 In a rhombus:
🔸 all four sides are equal
🔸 angles are not necessarily 90°
🔹 Construction of a rhombus:
🔹 Step 1: Draw a line segment AB of any length (for example 7 cm).
🔹 Step 2: With centre A and radius AB, draw an arc.
🔹 Step 3: With centre B and the same radius, draw another arc to cut the first arc at point C.
🔹 Step 4: With centre A and radius AB, draw another arc on the other side of AB.
🔹 Step 5: With centre B and the same radius, draw an arc to cut this arc at point D.
🔹 Step 6: Join AC, CB, BD, and DA.
🔹 All sides AB, BC, CD, and DA are equal.
🔹 The angles are not right angles.
🔹 Hence, the figure is a rhombus, not a square.
——————————————————————————————————————————————————————————————————————————–
OTHER IMPORTANT QUESTIONS
(CBSE MODEL QUESTION PAPER)
ESPECIALLY MADE FROM THIS CHAPTER ONLY
🔵 Section A — Very Short Answer
(Q1–Q6 | 1 × 6 = 6 marks)
🔵 Question
Q1. What is meant by a geometrical construction?
🟢 Answer
✔️ A geometrical construction is an accurate drawing made using a ruler and a compass by following fixed rules.
🔵 Question
Q2. Name the two main instruments used in geometrical constructions.
🟢 Answer
✔️ The two main instruments are a ruler (straightedge) and a compass.
🔵 Question
Q3. Is a ruler used for measuring lengths in constructions? (Yes/No)
🟢 Answer
✔️ No, a ruler is used only to draw straight lines, not for measuring.
🔵 Question
Q4. What is the fixed point of a circle called?
🟢 Answer
✔️ The fixed point of a circle is called its centre.
🔵 Question
Q5. What angle is formed by a perpendicular line?
🟢 Answer
✔️ A perpendicular line forms a right angle (90°).
🔵 Question
Q6. What does a perpendicular bisector do to a line segment?
🟢 Answer
✔️ A perpendicular bisector divides a line segment into two equal parts at right angles.
🟢 Section B — Short Answer–I
(Q7–Q12 | 2 × 6 = 12 marks)
🟢 Question
Q7. Why are geometrical constructions more accurate than freehand drawings?
🟢 Answer
🔵 Geometrical constructions use fixed instruments like a ruler and compass.
🔵 They follow logical steps, so lengths and angles are exact and not guessed.
🟢 Question
Q8. Write two uses of a compass in geometrical constructions.
🟢 Answer
🔵 A compass is used to draw circles and arcs.
🔵 It is also used to copy equal lengths and angles accurately.
🟢 Question
Q9. What is meant by copying a line segment?
🟢 Answer
🔵 Copying a line segment means drawing another line segment of exactly the same length using a compass.
🟢 Question
Q10. Define a perpendicular bisector.
🟢 Answer
🔵 A perpendicular bisector is a line that cuts a line segment into two equal halves and makes a right angle with it.
🟢 Question
Q11. What precaution should be taken while using a compass in constructions?
🟢 Answer
🔵 The opening of the compass should not be changed while copying a length or an angle.
🟢 Question
Q12. Why should construction lines be drawn lightly?
🟢 Answer
🔵 Light lines help in seeing intersections clearly and keep the construction neat and accurate.
🟡 Section C — Short Answer–II
(Q13–Q22 | 3 × 10 = 30 marks)
🟡 Question
Q13. What is the role of accuracy in geometrical constructions? Give three points.
🟢 Answer
🔵 Accurate constructions ensure correct lengths and angles.
🔵 Small errors can change the final figure completely.
🔵 Accuracy helps in understanding properties of shapes clearly.
🟡 Question
Q14. Explain how a line segment of a given length is constructed.
🟢 Answer
🔵 Draw a straight line using a ruler.
🔵 Mark a point on the line as the starting point.
🔵 Open the compass to the given length and mark the point on the line.
🟡 Question
Q15. What is a perpendicular bisector? State two of its properties.
🟢 Answer
🔵 A perpendicular bisector divides a line segment into two equal parts.
🔵 It makes a right angle (90°) with the line segment.
🔵 Every point on it is equidistant from the endpoints of the segment.
🟡 Question
Q16. Explain the steps to construct a circle of a given radius.
🟢 Answer
🔵 Mark the centre point on the paper.
🔵 Open the compass to the given radius.
🔵 Rotate the compass fully to draw the circle.
🟡 Question
Q17. Why is a ruler not used for measuring in geometrical constructions?
🟢 Answer
🔵 Measuring by a ruler may cause estimation errors.
🔵 Constructions require copying lengths exactly using a compass.
🔵 The ruler is used only to draw straight lines.
🟡 Question
Q18. What is meant by copying an angle? Why is the compass opening kept fixed?
🟢 Answer
🔵 Copying an angle means constructing another angle equal to a given angle.
🔵 Keeping the compass opening fixed ensures the new angle is exactly equal to the original angle.
🟡 Question
Q19. Explain how a perpendicular is drawn to a line from a point on it.
🟢 Answer
🔵 Take a point on the given line.
🔵 Draw arcs on both sides of the point.
🔵 From these points, draw arcs that intersect and join the intersection to the given point.
🟡 Question
Q20. Explain how a perpendicular is drawn to a line from a point outside it.
🟢 Answer
🔵 Draw arcs from the given point to cut the line at two points.
🔵 From these points, draw arcs that intersect above the line.
🔵 Join the intersection point to the given point.
🟡 Question
Q21. Write three precautions that should be taken while doing constructions.
🟢 Answer
🔵 Do not change the compass opening unnecessarily.
🔵 Draw construction lines lightly and neatly.
🔵 Keep the ruler and compass steady while drawing.
🟡 Question
Q22. How do geometrical constructions help in real life? Explain briefly.
🟢 Answer
🔵 They are used in architecture and building design.
🔵 Engineers use constructions for accurate drawings.
🔵 Map making and planning also depend on constructions.
🔴 Section D — Long Answer
(Q23–Q30 | 4 × 8 = 32 marks)
🔴 Question
Q23. Explain, with steps, how to construct a line segment of a given length.
🟢 Answer
🔵 Step 1: Draw a straight line using a ruler.
🔵 Step 2: Mark a point on the line as the starting point.
🔵 Step 3: Open the compass to the given length.
🔵 Step 4: With the needle at the starting point, mark the required point on the line.
✔️ The distance between the two marked points is the required line segment.
🔴 Question
Q24. Describe the construction of a circle of a given radius.
🟢 Answer
🔵 Step 1: Mark a point as the centre of the circle.
🔵 Step 2: Open the compass to the given radius.
🔵 Step 3: Place the needle at the centre and rotate the compass fully.
✔️ A circle with the given radius is obtained.
🔴 Question
Q25. Explain how to construct the perpendicular bisector of a given line segment.
🟢 Answer
🔵 Step 1: Draw the given line segment.
🔵 Step 2: Open the compass to more than half the length of the segment.
🔵 Step 3: With each endpoint as centre, draw arcs on both sides of the segment.
🔵 Step 4: Join the points where the arcs intersect.
✔️ The drawn line cuts the segment into two equal parts at right angles.
🔴 Question
Q26. Describe the steps to construct an angle equal to a given angle.
🟢 Answer
🔵 Step 1: Draw a ray for the new angle.
🔵 Step 2: Draw an arc from the vertex of the given angle to cut its arms.
🔵 Step 3: Draw the same arc on the new ray.
🔵 Step 4: Copy the distance between the arc points and join to form the angle.
✔️ The new angle is equal to the given angle.
🔴 Question
Q27. Explain the construction of a perpendicular to a line from a point on it.
🟢 Answer
🔵 Step 1: Take the given point on the line.
🔵 Step 2: With this point as centre, draw an arc to cut the line on both sides.
🔵 Step 3: With these cut points as centres, draw arcs that intersect.
🔵 Step 4: Join the intersection point with the given point.
✔️ The line drawn is perpendicular to the given line.
🔴 Question
Q28. Explain the construction of a perpendicular to a line from a point outside it.
🟢 Answer
🔵 Step 1: With the given point as centre, draw an arc cutting the line at two points.
🔵 Step 2: With these two points as centres, draw arcs that intersect above the line.
🔵 Step 3: Join the intersection point with the given point.
✔️ The constructed line is perpendicular to the given line.
🔴 Question
Q29. Why is accuracy important in geometrical constructions? Explain with examples.
🟢 Answer
🔵 Accurate constructions ensure correct lengths and angles.
🔵 Even a small change in compass width can change the whole figure.
🔵 Correct constructions help in understanding properties of shapes clearly.
🔵 Inaccurate drawings can lead to wrong conclusions in geometry problems.
✔️ Accuracy is essential for correct understanding and correct answers.
🔴 Question
Q30. Write four precautions that should be taken while doing geometrical constructions.
🟢 Answer
🔵 Do not change the compass opening while copying lengths or angles.
🔵 Draw construction lines lightly so intersections are visible.
🔵 Keep the ruler and compass steady while drawing.
🔵 Avoid guessing or measuring with a ruler during constructions.
✔️ Following these precautions ensures neat and accurate constructions.
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