Practical Geometry

Class 6 CBSE Maths Notes: Practical Geometry - Complete Guide

Introduction to Practical Geometry

Practical Geometry is a fundamental chapter in Class 6 CBSE Maths notes that introduces students to the art and science of geometric constructions. Unlike theoretical geometry that focuses on properties and theorems, practical geometry teaches students how to create accurate geometric figures using basic drawing instruments primarily a ruler (straight edge), compass, protractor, and set-square.

This chapter builds essential skills that form the foundation for advanced geometric concepts in higher classes. Students learn to construct line segments, angles, perpendiculars, parallels, and eventually more complex shapes like triangles and quadrilaterals with precision and accuracy.

Importance of Learning Practical Geometry

• Develops spatial reasoning and visualization skills
• Enhances precision and accuracy in mathematical work
• Provides hands-on experience with mathematical concepts
• Forms the foundation for technical drawing and engineering concepts
• Improves problem-solving abilities through systematic construction methods

Essential Tools for Practical Geometry

Before starting geometric constructions, familiarize yourself with these fundamental tools:

1. Ruler (Straight Edge)

• Used for drawing straight lines and measuring lengths
• Typically marked in centimeters and millimeters
• The zero mark should be clearly visible and accurate

2. Compass

• Consists of two legs one with a pointed end and one with a pencil
• Used for drawing arcs and circles
• Essential for transferring measurements and creating equal distances

3. Protractor

• A semicircular or circular instrument marked with degrees (0° to 180° or 0° to 360°)
• Used for measuring and constructing angles
• The center point and zero line must be aligned carefully

4. Set-Square

• Triangular instruments with specific angles (usually 45°-45°-90° and 30°-60°-90°)
• Used for drawing perpendicular and parallel lines
• Helpful for creating right angles quickly

Construction of Line Segments

Method 1: Using a Ruler

Steps:

1. Mark a point 'A' on the paper
2. Place the ruler so that its zero mark coincides with point A
3. Mark another point 'B' at the required length on the ruler
4. Join A and B with a pencil

Example: To construct a line segment of 5.3 cm, mark point A, align the ruler's zero with A, mark point B at the 5.3 cm mark, and join them.

Method 2: Using Compass and Ruler

This method is more accurate for transferring measurements:

Steps:

1. Mark a point A and draw a line passing through it
2. Place the metal point of the compass on the zero mark of the ruler
3. Open the compass to match the required length on the ruler
4. Without changing the compass width, place its metal point on A
5. Make a small arc on the line to mark point B
6. The segment AB is your required line segment

Practical Tip: Always keep the compass tight so the width doesn't change during construction.

Bisection of a Line Segment

Bisecting a line segment means dividing it into two equal parts. This construction also creates a perpendicular bisector.

Steps of Construction:

1. Draw the line segment AB of the given length
2. With center A and radius more than half of AB, draw arcs on both sides of AB
3. With center B and the same radius, draw arcs to intersect the previous arcs
4. Name the intersection points as E and F
5. Join E and F with a straight line
6. Mark the intersection point of AB and EF as M

Result: M is the midpoint of AB, and EF is the perpendicular bisector.

Why this works: The construction creates two congruent triangles (AEF and BEF), ensuring M is equidistant from both A and B.

Construction of Perpendicular Lines

Type 1: Perpendicular to a Line Through a Point ON the Line

Method A: Using Ruler and Set-Square

Steps:

1. Place a ruler with one edge along line l
2. Hold the ruler firmly in place
3. Place a set-square with one edge along the ruler, ensuring the right-angled corner touches the ruler
4. Slide the set-square along the ruler until its right-angled corner reaches point P
5. Draw along the edge of the set-square through P

Advantage: This method is quick and suitable for rough constructions.

Method B: Using Ruler and Compass (More Accurate)

Steps:

1. Given: A point P on line l
2. With P as center and a convenient radius, draw an arc intersecting line l at points A and B
3. With A and B as centers and radius greater than AP, draw two arcs intersecting at point Q
4. Join PQ
5. Result: PQ ⊥ l (PQ is perpendicular to l)

Why this works: The construction ensures that P is equidistant from A and B, and Q is also equidistant from A and B, creating a perpendicular bisector.

Type 2: Perpendicular to a Line Through a Point NOT on the Line

Method A: Using Ruler and Set-Square

Steps:

1. Place a set-square on line l so one arm of its right angle aligns with l
2. Place a ruler along the opposite edge of the set-square
3. Hold the ruler fixed and slide the set-square along it
4. Continue sliding until point P touches the other arm of the set-square
5. Draw along the edge through P, meeting l at M

Result: PM ⊥ l

Method B: Using Ruler and Compass

Steps:

1. Given: Line l and point P not on it
2. With P as center, draw an arc intersecting line l at two points A and B
3. With A and B as centers and the same radius, draw two arcs intersecting at point Q (on the opposite side of the line from P)
4. Join PQ

Result: PQ is perpendicular to l

Construction of Angles

Method 1: Constructing an Angle of Given Measure (Using Protractor)

Example: To construct a 40° angle

Steps:

1. Draw a line segment AB of any length
2. Place the center of the protractor at point A
3. Align the zero edge along AB
4. Mark point C at the 40° marking
5. Join AC

Result: ∠BAC = 40°

Method 2: Copying an Angle of Unknown Measure (Using Compass)

When you need to copy an angle without knowing its measurement:

Given: ∠A (measure unknown)

Steps:

1. Draw a line l and choose a point P on it
2. With A as center, draw an arc cutting the rays of ∠A at B and C
3. With P as center and the same radius, draw an arc cutting l at Q
4. Measure BC with the compass (set compass width to BC)
5. With Q as center and radius equal to BC, draw an arc intersecting the previous arc at R
6. Join PR

Result: ∠QPR has the same measure as ∠BAC

Application: This method is useful when you need to transfer an angle from one location to another in a construction.

Construction of Special Angles

Constructing a 60° Angle

A 60° angle is fundamental as it's one-third of a straight angle and forms part of an equilateral triangle.

Steps:

1. Draw a line l and mark a point O on it
2. With O as center and convenient radius, draw an arc cutting the line at point A
3. With A as center and the same radius, draw an arc passing through O
4. Let the two arcs intersect at B
5. Join OB

Result: ∠BOA = 60°

Why this works: The three points O, A, and B form an equilateral triangle where all angles are 60°.

Constructing a 120° Angle

A 120° angle is twice a 60° angle.

Steps:

1. Draw line PQ and take point O on it
2. With O as center, draw an arc cutting the line at A
3. With A as center and the same radius, draw an arc cutting the first arc at B
4. With B as center and the same radius, draw another arc cutting the first arc at C
5. Join OC

Result: ∠COA = 120°

Explanation: Each step creates a 60° angle, so two steps create 120°.

Constructing a 90° Angle (Right Angle)

Method 1: Using 60° and 120° Angles

Steps:

1. Construct ∠AOB = 60°
2. Construct ∠AOF = 120°
3. Bisect the angle ∠BOF
4. Let OP be the bisector

Result: ∠AOP = 90°

Calculation: The angle between the 60° and 120° rays is 60°. Bisecting this gives 30°, so 60° + 30° = 90°.

Method 2: Using Set-Square Simply place the right angle of a set-square at the desired point.

Constructing a 30° Angle

Method: Bisect a 60° angle

Constructing a 45° Angle

Method: Bisect a 90° angle

Constructing a 75° Angle

Method: Add a 30° angle to a 45° angle, or construct 60° + 15° (where 15° is half of 30°)

Bisection of an Angle

An angle bisector divides an angle into two equal parts.

Given: ∠A

Steps:

1. With A as center, draw an arc that cuts both rays of ∠A at points B and C
2. With B as center and radius more than half of BC, draw an arc in the interior of ∠A
3. With C as center and the same radius, draw another arc intersecting the previous arc at D
4. Join AD

Result: AD is the bisector of ∠A

Verification: ∠BAD = ∠CAD

How to Construct Parallel Lines

Parallel lines are lines that never meet, no matter how far they are extended. They maintain a constant distance from each other.

Construction of Parallel Line Through a Given Point

Given: Line AB and point P outside AB

Steps:

1. Take any point Q on line AB
2. Join P and Q
3. With Q as center and convenient radius, draw an arc intersecting AB at C and PQ at D
4. With P as center and the same radius, draw an arc intersecting PQ at E
5. With E as center and radius equal to CD, draw an arc cutting the previous arc at F
6. Join PF and extend in both directions

Result: Line PF is parallel to AB

Principle: This construction creates equal corresponding angles, which is a condition for parallel lines.

Alternative Method: Using Set-Square and Ruler

Steps:

1. Place a ruler along line AB
2. Place a set-square with one edge along AB
3. Slide the set-square along the ruler to point P
4. Draw a line along the edge of the set-square through P

This method is quicker but less precise than the compass method.

Construction of Triangles

A triangle is a polygon with three sides and three angles. The sum of all angles in a triangle is always 180°.

Triangle Construction Methods

There are four main criteria for constructing triangles:

1. SSS (Side-Side-Side) Method

When Given: All three sides of the triangle

Example: Construct triangle ABC with AB = 5 cm, BC = 6 cm, AC = 4 cm

Steps:

1. Draw a line segment BC = 6 cm
2. With B as center and radius 5 cm, draw an arc above BC
3. With C as center and radius 4 cm, draw another arc intersecting the first arc at A
4. Join AB and AC

Result: Triangle ABC is constructed

Important Note: The sum of any two sides must be greater than the third side, otherwise the triangle cannot be constructed.

2. SAS (Side-Angle-Side) Method

When Given: Two sides and the included angle (angle between those sides)

Example: Construct triangle PQR with PQ = 4 cm, ∠QPR = 60°, PR = 5 cm

Steps:

1. Draw a line segment PQ = 4 cm
2. At point P, construct ∠QPR = 60°
3. On ray PR, mark point R such that PR = 5 cm
4. Join QR

Result: Triangle PQR is constructed

3. ASA (Angle-Side-Angle) Method

When Given: Two angles and the included side (side between those angles)

Example: Construct triangle XYZ with ∠Y = 45°, YZ = 5.5 cm, ∠Z = 60°

Steps:

1. Draw a line segment YZ = 5.5 cm
2. At point Y, construct ∠XYZ = 45°
3. At point Z, construct ∠XZY = 60°
4. Let the two rays intersect at X

Result: Triangle XYZ is constructed

Important Note: The sum of the two given angles must be less than 180°.

4. RHS (Right angle-Hypotenuse-Side) Method

When Given: The hypotenuse and one side of a right-angled triangle

Example: Construct right triangle ABC with ∠B = 90°, AC = 6 cm, AB = 4 cm

Steps:

1. Draw a line segment AB = 4 cm
2. At point B, construct a 90° angle
3. With A as center and radius 6 cm, draw an arc cutting the perpendicular at C
4. Join AC

Result: Right triangle ABC is constructed

Verification: Use Pythagoras theorem: AC² = AB² + BC²

Construction of Quadrilaterals

A quadrilateral is a four-sided polygon. To construct a quadrilateral uniquely, you need five measurements (sides and angles).

Common Quadrilateral Constructions

Method 1: When Four Sides and One Diagonal are Given

Example: Construct quadrilateral ABCD with AB = 4 cm, BC = 3 cm, CD = 5 cm, DA = 5.5 cm, and diagonal AC = 6 cm

Steps:

1. Construct triangle ABC using SSS method with AB = 4 cm, BC = 3 cm, AC = 6 cm
2. With C as center and radius 5 cm, draw an arc on the opposite side of B
3. With A as center and radius 5.5 cm, draw an arc intersecting the previous arc at D
4. Join CD and AD

Result: Quadrilateral ABCD is constructed

Method 2: When Three Sides and Two Included Angles are Given

Example: Construct quadrilateral PQRS with PQ = 3.5 cm, ∠P = 75°, PS = 4 cm, ∠S = 105°, SR = 5 cm

Steps:

1. Draw PQ = 3.5 cm
2. At P, construct ∠P = 75° and mark PS = 4 cm on this ray
3. At S, construct ∠S = 105° and mark SR = 5 cm on this ray
4. Join QR

Result: Quadrilateral PQRS is constructed

Method 3: Special Quadrilaterals

For Rectangles:

• All angles are 90°
• Opposite sides are equal
• Need only length and breadth

For Squares:

• All angles are 90°
• All sides are equal
• Need only one side length

For Parallelograms:

• Opposite sides are parallel and equal
• Opposite angles are equal
• Need two adjacent sides and one angle

Common Compass and Ruler Exercises for Class 6-8

Basic Level (Class 6)

1. Construct a line segment of length 7.3 cm
2. Bisect a line segment of 6 cm
3. Construct angles: 30°, 45°, 60°, 90°, 120°
4. Draw perpendicular to a line through a point on it
5. Copy a given angle using compass
6. Construct an equilateral triangle with side 5 cm

Intermediate Level (Class 7)

1. Construct perpendicular to a line from a point not on it
2. Draw parallel lines using compass and ruler
3. Construct triangles using SSS, SAS, ASA methods
4. Bisect various angles (acute, obtuse, right)
5. Construct a triangle given base, base angle, and sum of other two sides
6. Divide a line segment in given ratio (e.g., 2:3)

Advanced Level (Class 8)

1. Construct quadrilaterals with various given conditions
2. Construct special quadrilaterals (square, rectangle, rhombus, parallelogram)
3. Construct a triangle given perimeter and two angles
4. Construct tangents to circles
5. Construct angle bisectors and medians of triangles
6. Construct right triangles using RHS criterion

Printable Practice Problems and Solutions

Practice Set 1: Line Segments and Perpendiculars

Problem 1: Draw a line segment AB of length 8 cm. Locate point C on AB such that AC:BC = 1:3.

Solution:

1. Draw AB = 8 cm
2. Draw perpendicular bisector of AB (gives midpoint O where AO = 4 cm)
3. Draw perpendicular bisector of AO (gives point C where AC = 2 cm)
4. Now AC:BC = 2:6 = 1:3

Problem 2: Construct a perpendicular to line segment PQ of length 4.9 cm from point R not on PQ.

Solution:

1. Draw PQ = 4.9 cm
2. Mark point R outside PQ
3. With R as center, draw an arc cutting PQ at two points A and B
4. With A and B as centers and equal radii (more than half AB), draw arcs intersecting at S
5. Join RS (This is perpendicular to PQ)

Practice Set 2: Angles

Problem 3: Construct ∠CAB = 30° and ∠ABD = 45° with AB = 5 cm.

Solution:

1. Draw AB = 5 cm
2. At A, construct 60° angle and bisect it to get 30° angle (ray AC)
3. At B, construct 90° angle and bisect it to get 45° angle (ray BD)

Problem 4: Construct an angle of 75° and bisect it.

Solution:

1. Construct 60° angle
2. Construct 90° angle from same vertex
3. Bisect the 30° angle between them to get 15°
4. 60° + 15° = 75°
5. Bisect the 75° angle to get two 37.5° angles

Practice Set 3: Triangles

Problem 5: Construct an equilateral triangle ABC with side 5 cm. Draw the internal angle bisectors of ∠A and ∠C.

Solution:

1. Draw BC = 5 cm
2. With B and C as centers and radius 5 cm, draw arcs intersecting at A
3. Join AB and AC (equilateral triangle formed)
4. To bisect ∠A: Draw an arc from A cutting AB and AC, then from these points draw arcs intersecting inside, join to A
5. Repeat for ∠C

Problem 6: Construct triangle PQR with PQ = 6.4 cm, ∠RPQ = 60°, and ∠PQR = 45°. Draw a line through R parallel to PQ.

Solution:

1. Draw PQ = 6.4 cm
2. At P, construct 60° angle
3. At Q, construct 45° angle
4. Let the rays intersect at R
5. To draw parallel: Take point on PQ, join to R, copy the angle at R to create parallel line

Formulas and Properties

Formula/Property	Mathematical Representation	Explanation
Sum of angles in triangle	∠A + ∠B + ∠C = 180°	The three interior angles of any triangle always sum to 180°
Triangle inequality	AB + BC > AC	Sum of any two sides must be greater than the third side
Perpendicular lines	If l₁ ⊥ l₂, then angle = 90°	Perpendicular lines meet at right angles
Parallel lines property	If l₁ ∥ l₂, corresponding angles are equal	When a transversal cuts parallel lines, corresponding angles are equal
Angle bisector	If AD bisects ∠BAC, then ∠BAD = ∠CAD	Angle bisector divides angle into two equal parts
Perpendicular bisector	If PQ bisects AB at M, then AM = MB and PQ ⊥ AB	Perpendicular bisector divides segment into equal parts at 90°
Equilateral triangle	All sides equal, all angles = 60°	AB = BC = CA and ∠A = ∠B = ∠C = 60°
Isosceles triangle	Two sides equal, two angles equal	If AB = AC, then ∠B = ∠C
Right triangle (Pythagoras)	AC² = AB² + BC²	In right triangle with right angle at B, hypotenuse² = sum of squares of other sides
Sum of angles in quadrilateral	∠A + ∠B + ∠C + ∠D = 360°	The four interior angles of any quadrilateral sum to 360°

Common Mistakes to Avoid

1. Compass Width Changes

Mistake: Not keeping compass tight, causing radius to change during construction

Solution: Always check compass tightness before and during construction

2. Incorrect Protractor Alignment

Mistake: Not aligning the protractor's center and zero line properly

Solution: Double-check alignment before marking angles

3. Light Pencil Marks

Mistake: Using too much pressure, making marks difficult to erase or adjust

Solution: Use light, consistent pressure; construction lines can be darkened later

4. Ignoring Triangle Inequality

Mistake: Attempting to construct triangles where sum of two sides ≤ third side

Solution: Always verify the triangle inequality before construction

5. Not Maintaining Arc Radius

Mistake: Changing compass width when drawing multiple arcs

Solution: Mark the required radius on scrap paper or verify against ruler

6. Poor Point Marking

Mistake: Making thick or unclear intersection points

Solution: Use sharp pencil and mark points with small, clear dots

Tips for Accurate Constructions

1. Use a sharp pencil (HB or 2H grade works best)
2. Keep instruments clean and in good condition
3. Work on smooth, flat surface with proper lighting
4. Draw light construction lines that can be erased
5. Label all points clearly as you construct
6. Verify measurements at each step
7. Use set-squares for quick right angles
8. Practice regularly to improve accuracy
9. Understand the logic behind each construction
10. Check final figure against given measurements

Conclusion

Practical Geometry is a foundational chapter that develops crucial skills for mathematical thinking and precision. The constructions learned in Class 6 CBSE form the basis for more advanced geometric concepts in higher classes. By mastering these fundamental techniques constructing line segments, angles, perpendiculars, parallels, and basic shapes students build confidence in their ability to work with geometric concepts both theoretically and practically.

Takeaways:

• Always use proper instruments and maintain them well
• Understand the logic behind each construction, not just the steps
• Practice regularly to improve accuracy and speed
• Verify measurements at each step
• Light construction lines can always be darkened or erased

Resources for Further Practice

1. NCERT Textbook: Complete all exercise problems
2. NCERT Exemplar: For challenging problems
3. Previous year CBSE papers: For exam pattern familiarity
4. Online geometry tools: For visualization (GeoGebra, etc.)
5. Practice worksheets: Available on educational websites

Study Tips for Exam Success

• Create a construction toolkit and keep it ready
• Practice at least 5-10 constructions daily
• Time yourself to improve speed
• Understand the reasoning behind each step
• Revise different construction methods regularly
• Label all diagrams clearly in exams
• Show all construction arcs and marks

With consistent practice and understanding of fundamental principles, practical geometry becomes an enjoyable and rewarding topic. The skills learned here extend beyond mathematics into fields like engineering, architecture, design, and various sciences.

Author's Note: These notes have been compiled based on the CBSE Class 6 Mathematics curriculum and standard textbooks. The explanations combine theoretical understanding with practical, step-by-step guidance to help students master geometric constructions effectively.