Notes For All Chapters – Maths Class 6 Ganita Prakash
Class 6 · Mathematics · Chapter 9
📐 Playing with Constructions
Learn to draw accurate geometric figures using a ruler and compass — the ancient art of construction!
📏 Ruler & Compass
🔵 Circles
📐 Perpendiculars
🔷 Squares & Rectangles
✏️ NCERT Grade 6
🔵 Circles
📐 Perpendiculars
🔷 Squares & Rectangles
✏️ NCERT Grade 6
Contents
9.1 Introduction – What is Construction?
In geometry, construction means drawing accurate figures using only specific tools — a ruler (straight edge) and a compass — without measuring angles with a protractor.
Did You Know?
Ancient Greek mathematicians like Euclid used only a ruler and compass to construct perfect geometric figures. This tradition is thousands of years old and forms the foundation of geometry!
Ancient Greek mathematicians like Euclid used only a ruler and compass to construct perfect geometric figures. This tradition is thousands of years old and forms the foundation of geometry!
🎯 Why Learn Constructions?
- To draw perfectly accurate geometric shapes without guessing
- Used in engineering, architecture, and design
- Develops logical and precise thinking
- Helps understand geometric properties deeply
📌 Key Idea
When we construct a figure, we follow a set of steps. Each step uses either a ruler (to draw straight lines) or a compass (to draw arcs and circles). The result is an exact, accurate figure.
Construction = Drawing accurate figures using ONLY a Ruler + Compass (no measuring angles!)
9.2 Tools for Construction
🔧 The Two Main Tools
📏 Ruler (Straight Edge)
Used to draw straight lines and line segments. In construction, we use it only to draw lines — NOT to measure lengths with the markings.
Used to draw straight lines and line segments. In construction, we use it only to draw lines — NOT to measure lengths with the markings.
🔵 Compass
Used to draw circles and arcs. You can set any radius you like and draw perfect circular arcs. This is the most important construction tool.
Used to draw circles and arcs. You can set any radius you like and draw perfect circular arcs. This is the most important construction tool.
📎 Other Helpful Tools
- Pencil — always use a sharp pencil for accuracy
- Eraser — to clean up extra marks
- Protractor — used to measure angles (not used in pure construction)
- Set squares — triangular rulers with 90°, 45°, 60° angles
Pro Tip
Always keep your pencil sharp and apply light pressure. Construction marks should be thin and neat. Label all points clearly with capital letters like A, B, C.
Always keep your pencil sharp and apply light pressure. Construction marks should be thin and neat. Label all points clearly with capital letters like A, B, C.
Common Mistake
Don’t press the compass too hard — the pin point might slip, making your arcs inaccurate. Hold it steady from the top with gentle, consistent pressure.
Don’t press the compass too hard — the pin point might slip, making your arcs inaccurate. Hold it steady from the top with gentle, consistent pressure.
9.3 Drawing Circles
A circle is the set of all points that are the same distance from a fixed point called the centre. The fixed distance is called the radius.
🔴 Centre
The fixed middle point of a circle. Usually labelled O.
The fixed middle point of a circle. Usually labelled O.
📏 Radius
The distance from the centre to any point on the circle. All radii of a circle are equal.
The distance from the centre to any point on the circle. All radii of a circle are equal.
📐 How to Draw a Circle with Compass
- Mark the centre point O on your paper.
- Open your compass to the required radius (e.g., 3 cm) using a ruler.
- Place the metal pin of the compass exactly on point O.
- Hold the compass from the top and rotate it fully to draw the circle.
- The pencil traces a perfect circle of the given radius.
Remember
The diameter of a circle = 2 × radius. If radius = 4 cm, then diameter = 8 cm. The diameter is the longest chord and passes through the centre.
The diameter of a circle = 2 × radius. If radius = 4 cm, then diameter = 8 cm. The diameter is the longest chord and passes through the centre.
🎨 Interesting Patterns with Circles
By drawing circles of different radii from the same centre (called concentric circles), or by placing the compass pin on the circumference and drawing arcs, we can make beautiful patterns!
Concentric Circles = Multiple circles with the SAME centre but different radii
🔄 Intersecting Circles
When two circles overlap, they create interesting shapes. The points where two circles cross are called points of intersection. This idea is used in many beautiful geometric designs.
🌸
Fun with Circles!
The “Flower of Life” pattern — found in ancient temples including those in India — is made entirely by drawing circles of equal radius where each circle passes through the centre of the neighbouring circle!
The “Flower of Life” pattern — found in ancient temples including those in India — is made entirely by drawing circles of equal radius where each circle passes through the centre of the neighbouring circle!
9.4 Perpendicular Lines
Two lines (or line segments) are perpendicular to each other if they meet at a right angle (90°).
Perpendicular Lines → Meet at exactly 90° (a right angle)
📏 Drawing a Perpendicular Using a Set Square
- Draw a line AB using a ruler.
- Mark the point P on the line where you want the perpendicular.
- Place one edge of the set square along line AB, so the right angle corner is at P.
- Draw a line along the other edge of the set square through P.
- This new line is perpendicular to AB at point P.
🔵 Drawing a Perpendicular Using Compass
Perpendicular from an external point P to a line AB:
- With P as centre, draw an arc that cuts line AB at two points — call them C and D.
- With C as centre and a radius more than half of CD, draw an arc below the line.
- With D as centre and the same radius, draw another arc to intersect the first arc. Call this point Q.
- Join P and Q. The line PQ is perpendicular to AB.
Key Idea
The foot of the perpendicular is the point where the perpendicular meets the line. The perpendicular distance from a point to a line is the shortest distance between them.
The foot of the perpendicular is the point where the perpendicular meets the line. The perpendicular distance from a point to a line is the shortest distance between them.
⊥ Symbol
We write “PQ ⊥ AB” to mean “PQ is perpendicular to AB”. The small square □ at the intersection shows a 90° angle.
We write “PQ ⊥ AB” to mean “PQ is perpendicular to AB”. The small square □ at the intersection shows a 90° angle.
📐 Right Angle
A right angle measures exactly 90°. In constructions, a small square is always drawn at the corner to mark a right angle.
A right angle measures exactly 90°. In constructions, a small square is always drawn at the corner to mark a right angle.
9.5 Perpendicular Bisector of a Line Segment
The perpendicular bisector of a line segment is a line that:
- Cuts the line segment into two equal halves (bisects it)
- Is perpendicular (at 90°) to the line segment
Perpendicular Bisector = Passes through Midpoint + Makes 90° angle with the segment
🔵 Steps to Draw Perpendicular Bisector of Line Segment AB
- Draw a line segment AB of given length (say 6 cm).
- Open the compass to a radius more than half of AB (more than 3 cm).
- With A as centre, draw arcs above and below the line segment.
- With B as centre and the same radius, draw arcs above and below, cutting the previous arcs at points P and Q.
- Join P and Q. This line PQ is the perpendicular bisector of AB.
- The point where PQ meets AB is the midpoint M of AB.
Important Property
Every point on the perpendicular bisector of a line segment is equidistant (equally far) from the two endpoints of the segment. This is a very useful geometric property!
Every point on the perpendicular bisector of a line segment is equidistant (equally far) from the two endpoints of the segment. This is a very useful geometric property!
If AB = 6 cm, and M is the midpoint found by perpendicular bisector:
AM = MB = 3 cm ← AB is bisected into two equal parts
Angle at M = 90° ← perpendicular condition
AM = MB = 3 cm ← AB is bisected into two equal parts
Angle at M = 90° ← perpendicular condition
Note
The perpendicular bisector is also called the “right bisector.” It is used in many constructions — finding the centre of a circle, constructing squares, and more.
The perpendicular bisector is also called the “right bisector.” It is used in many constructions — finding the centre of a circle, constructing squares, and more.
9.6 Angle Bisector
An angle bisector is a ray that divides an angle into two equal angles.
Angle Bisector divides ∠AOB into two equal angles: ∠AOC = ∠COB = ½ × ∠AOB
🔵 Steps to Bisect an Angle using Compass
Let’s bisect angle ∠AOB:
- With O as centre, draw an arc of any radius that cuts OA at point P and OB at point Q.
- With P as centre, draw an arc inside the angle (choose any suitable radius).
- With Q as centre and the same radius, draw another arc to intersect the previous arc at point R.
- Draw ray OR. This is the angle bisector of ∠AOB.
- Now ∠AOR = ∠BOR — both are equal!
Common Error
When drawing arcs from P and Q in steps 2–3, make sure you use the exact same radius for both arcs. If the radii are different, the two arcs won’t intersect correctly, and your bisector will be wrong.
When drawing arcs from P and Q in steps 2–3, make sure you use the exact same radius for both arcs. If the radii are different, the two arcs won’t intersect correctly, and your bisector will be wrong.
📐 Bisecting 90°
If we bisect a right angle (90°), each half = 45°. This is how we construct a 45° angle using compass.
If we bisect a right angle (90°), each half = 45°. This is how we construct a 45° angle using compass.
📐 Bisecting 60°
Bisecting a 60° angle gives two 30° angles. This is used to construct a 30° angle.
Bisecting a 60° angle gives two 30° angles. This is used to construct a 30° angle.
9.7 Constructing a Square
A square is a quadrilateral with:
- All four sides equal
- All four angles equal to 90°
- Diagonals that are equal in length and bisect each other at right angles
Square: All sides equal + All angles = 90°
🔵 Method 1: Constructing a Square Given One Side
Given: Side length = 4 cm. Construct square ABCD.
- Draw AB = 4 cm using a ruler.
- At point A, draw a perpendicular using compass or set square.
- On this perpendicular, mark point D such that AD = 4 cm.
- At point B, draw a perpendicular to AB. Mark point C such that BC = 4 cm.
- Join D and C. Verify that DC = 4 cm.
- ABCD is the required square. Check: all sides = 4 cm, all angles = 90°.
🔵 Method 2: Using Diagonals
We can also construct a square using the property that its diagonals bisect each other at right angles and are equal.
- Draw diagonal AC of the given length (say 6 cm).
- Find the midpoint O of AC using perpendicular bisector.
- At O, draw the perpendicular bisector of AC.
- On this bisector, mark points B and D on either side of O, each at distance AC/2 = 3 cm from O.
- Join A, B, C, D. This is the required square.
Verification Tip
After constructing, always verify your square: measure all 4 sides (should be equal), check all 4 angles with a protractor (should all be 90°), and measure both diagonals (should be equal).
After constructing, always verify your square: measure all 4 sides (should be equal), check all 4 angles with a protractor (should all be 90°), and measure both diagonals (should be equal).
🏛️
Squares in Architecture
Many ancient Indian temples and monuments were designed using compass-and-ruler constructions. The Sulbasutras (ancient Indian texts) contain detailed instructions for constructing squares and rectangles for building fire altars!
Many ancient Indian temples and monuments were designed using compass-and-ruler constructions. The Sulbasutras (ancient Indian texts) contain detailed instructions for constructing squares and rectangles for building fire altars!
9.8 Constructing a Rectangle
A rectangle is a quadrilateral with:
- Opposite sides equal and parallel
- All four angles equal to 90°
- Diagonals that are equal in length and bisect each other (but NOT at right angles)
Rectangle: Opposite sides equal + All angles = 90° (but adjacent sides can be different)
🔵 Steps to Construct Rectangle ABCD (length = 5 cm, width = 3 cm)
- Draw AB = 5 cm (the longer side = length).
- At point A, construct a perpendicular to AB.
- On this perpendicular, mark point D such that AD = 3 cm (the width).
- At point B, construct a perpendicular to AB.
- On this perpendicular, mark point C such that BC = 3 cm.
- Join D and C. Verify DC = 5 cm.
- ABCD is the required rectangle.
| Property | Square | Rectangle |
|---|---|---|
| All sides equal | ✅ Yes | ❌ No (only opposite sides equal) |
| All angles = 90° | ✅ Yes | ✅ Yes |
| Diagonals equal | ✅ Yes | ✅ Yes |
| Diagonals bisect at 90° | ✅ Yes | ❌ No |
| Is a Square a Rectangle? | ✅ Yes! Every square is a special rectangle. | |
Key Relationship
Every square is a rectangle (because it has all angles = 90° and opposite sides equal), but not every rectangle is a square (a rectangle’s adjacent sides can be unequal).
Every square is a rectangle (because it has all angles = 90° and opposite sides equal), but not every rectangle is a square (a rectangle’s adjacent sides can be unequal).
🔵 Using the Diagonal Property to Construct a Rectangle
Since a rectangle’s diagonals are equal and bisect each other:
- Draw diagonal AC (calculate using Pythagoras: AC = √(l² + b²)).
- Find midpoint O (perpendicular bisector of AC).
- Draw another diagonal BD = AC through O, but not necessarily perpendicular to AC.
- Connect A, B, C, D to form the rectangle.
Remember
The diagonal of a rectangle with length l and breadth b is d = √(l² + b²). For a 3 cm × 4 cm rectangle, the diagonal = √(9+16) = √25 = 5 cm!
The diagonal of a rectangle with length l and breadth b is d = √(l² + b²). For a 3 cm × 4 cm rectangle, the diagonal = √(9+16) = √25 = 5 cm!
Chapter Summary
🔵 Circle
Set of points equidistant from centre. Drawn with compass. Radius = distance from centre to any point on circle.
Set of points equidistant from centre. Drawn with compass. Radius = distance from centre to any point on circle.
📐 Perpendicular
Two lines meeting at 90°. Drawn using set square or compass. Marked with a small square symbol.
Two lines meeting at 90°. Drawn using set square or compass. Marked with a small square symbol.
✂️ Perp. Bisector
Divides a line segment into two equal parts at 90°. Every point on it is equidistant from both endpoints.
Divides a line segment into two equal parts at 90°. Every point on it is equidistant from both endpoints.
🔀 Angle Bisector
Divides an angle into two equal halves. Constructed using compass arcs from both sides of the angle.
Divides an angle into two equal halves. Constructed using compass arcs from both sides of the angle.
🔷 Square
All 4 sides equal + all angles 90°. Diagonals equal and bisect at 90°. Special rectangle.
All 4 sides equal + all angles 90°. Diagonals equal and bisect at 90°. Special rectangle.
▭ Rectangle
Opposite sides equal + all angles 90°. Diagonals equal and bisect each other. NOT necessarily at 90°.
Opposite sides equal + all angles 90°. Diagonals equal and bisect each other. NOT necessarily at 90°.
📏 Key Construction Steps — Quick Reference
| Construction | Steps in Brief | Key Tool |
|---|---|---|
| Circle of radius r | Set compass to r, pin at centre, rotate | Compass |
| Perpendicular at point P on line | Arc from P → intersects line at X, Y → arcs from X, Y → join P to intersection | Compass + Ruler |
| Perpendicular bisector of AB | Arcs from A and B (radius > AB/2) → join the two intersection points | Compass + Ruler |
| Angle bisector of ∠AOB | Arc from O → arcs from P and Q → join O to intersection | Compass + Ruler |
| Square of side a | Draw AB = a, perpendiculars at A and B, mark D and C at height a, join DC | Compass + Ruler |
| Rectangle l × b | Draw AB = l, perpendiculars at A and B, mark D and C at height b, join DC | Compass + Ruler |
Exam Practice Questions
🟢 1-mark Questions
Q1. Name the two basic tools used for geometric constructions.
Ans: A ruler (straight edge) and a compass.
Q2. What is a perpendicular bisector?
Ans: A perpendicular bisector is a line that divides a given line segment into two equal halves and is perpendicular (at 90°) to it.
Q3. What symbol is used to show two perpendicular lines?
Ans: The symbol ⊥ is used. For example, PQ ⊥ AB means PQ is perpendicular to AB.
Q4. If two circles have the same centre, what are they called?
Ans: They are called concentric circles.
🔵 2-mark Questions
Q5. What is the difference between a square and a rectangle? Give two points.
Ans: (i) In a square, all four sides are equal; in a rectangle, only opposite sides are equal. (ii) In a square, diagonals bisect each other at 90°; in a rectangle, diagonals do NOT bisect at 90°.
Q6. The radius of a circle is 5 cm. What is its diameter? If another concentric circle has a radius of 3 cm, what is the distance between the two circles at any point?
Ans: Diameter = 2 × 5 = 10 cm. Distance between the two circles = 5 − 3 = 2 cm (along any radius).
Q7. State the important property of all points on a perpendicular bisector of segment AB.
Ans: Every point on the perpendicular bisector of AB is equidistant from A and B. That is, if P is any point on the perpendicular bisector, then PA = PB.
🔴 3–4 mark Questions (Construction Steps)
Q8. Construct the perpendicular bisector of a line segment of 8 cm. Write all steps clearly.
Ans: Step 1: Draw segment AB = 8 cm. Step 2: With A as centre and radius > 4 cm, draw arcs above and below AB. Step 3: With B as centre, same radius, draw arcs cutting the previous ones at P and Q. Step 4: Join PQ. PQ is the perpendicular bisector, meeting AB at midpoint M. AM = MB = 4 cm, ∠PMA = 90°.
Q9. Construct a square of side 5 cm. Write the steps of construction.
Ans: Step 1: Draw AB = 5 cm. Step 2: Construct perpendicular at A; mark D on it with AD = 5 cm. Step 3: Construct perpendicular at B; mark C on it with BC = 5 cm. Step 4: Join DC. Verify DC = 5 cm. ABCD is the required square with all sides 5 cm and all angles 90°.
Q10. Is every square a rectangle? Is every rectangle a square? Justify your answer.
Ans: Yes, every square IS a rectangle, because a square has all angles equal to 90° and opposite sides equal — which satisfies the definition of a rectangle. But NOT every rectangle is a square, because a rectangle can have adjacent sides of different lengths (e.g., 5 cm × 3 cm), while a square must have all four sides equal.
Exam Strategy
For construction questions, always: (1) write numbered steps clearly, (2) draw a neat labelled figure, (3) verify your construction by measuring at the end, and (4) mention all measurements. Full marks depend on neat work and correct steps!
For construction questions, always: (1) write numbered steps clearly, (2) draw a neat labelled figure, (3) verify your construction by measuring at the end, and (4) mention all measurements. Full marks depend on neat work and correct steps!
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