Class 8 Mathematics Chapter 4: Practical Geometry 

Introduction :Practical Geometry

Practical Geometry deals with the construction of geometric figures using only a ruler (straight edge) and a pair of compasses. In Class 8, we learn to construct quadrilaterals when sufficient data (sides and angles/diagonals) is given. The chapter is based on the fact that a quadrilateral can be constructed uniquely if certain minimum measurements are known.

Tools Required

1. Ruler (scale)
2. Pair of compasses
3. Protractor (only for measuring angles when needed)
4. Pencil and eraser

Key Concept: Minimum Conditions for Unique Quadrilateral

To construct a unique quadrilateral, we need 5 independent measurements (sides, angles, diagonals). Common cases taught in Class 8:

The chapter focuses on the first four cases.

Construction 4.1: When Four Sides and One Diagonal are Given

Example: Construct quadrilateral ABCD such that AB = 4 cm, BC = 5 cm, CD = 4.5 cm, DA = 5.5 cm, AC = 6 cm.

Steps:

1. Draw side AB = 4 cm.
2. With A as centre and radius 6 cm, draw an arc.
3. With B as centre and radius 5 cm, draw another arc to intersect the previous arc at point C. (△ABC is constructed using SSS)
4. With C as centre and radius 4.5 cm, draw an arc.
5. With A as centre and radius 5.5 cm, draw another arc to intersect the previous arc at D.
6. Join CD and DA. Quadrilateral ABCD is ready.

Construction 4.2: When Three Sides and Two Diagonals are Given

Example: Construct quadrilateral ABCD where AB = 5 cm, BC = 4.5 cm, AC = 6 cm, BD = 7 cm, CD = 5 cm.

Steps:

1. Draw AB = 5 cm.
2. With A as centre, radius 6 cm draw arc.
3. With B as centre, radius 4.5 cm draw arc to get point C. (△ABC constructed)
4. With B as centre, radius 7 cm draw arc on the opposite side of A.
5. With C as centre, radius 5 cm draw another arc to intersect previous arc at D.
6. Join CD and BD. Quadrilateral ABCD is constructed.

(Note: The diagonal BD is used to locate D.)

Construction 4.3: When Three Sides and Two Included Angles are Given

Example: Construct quadrilateral ABCD where AB = 5 cm, BC = 4 cm, CD = 5.5 cm, ∠ABC = 80°, ∠BCD = 110°.

Steps:

1. Draw BC = 4 cm.
2. At B, construct ∠ABC = 80°. Along this ray, mark AB = 5 cm.
3. At C, construct ∠BCD = 110°. Draw the ray.
4. From A, draw an arc of radius 5.5 cm (CD length).
5. From the ray drawn from C, the intersection point of arc and ray is D.
6. Join AD. Quadrilateral ABCD is ready.

Construction 4.4: When Four Sides and One Angle are Given

Example: Construct quadrilateral PQRS where PQ = 4 cm, QR = 5 cm, RS = 4.5 cm, SP = 6 cm, ∠PQR = 75°.

Steps:

1. Draw QR = 5 cm.
2. At Q, construct ∠PQR = 75°. Along this ray mark PQ = 4 cm.
3. From P, draw arc of radius SP = 6 cm.
4. From R, draw arc of radius RS = 4.5 cm.
5. The intersection point of both arcs is S.
6. Join RS and SP. Quadrilateral PQRS is constructed.

Special Quadrilaterals (Rhombus, Square, Rectangle, Parallelogram)

1. Rhombus

All sides equal.

To construct a rhombus when: (a) Side and one angle given (b) Two diagonals given (easiest)

Construction using diagonals: Diagonals of a rhombus bisect each other at 90°.

Steps:

1. Draw diagonal AC.
2. Find midpoint O of AC.
3. Draw perpendicular bisector of AC through O.
4. From O, cut off OB = OD = half of second diagonal.
5. Join ABCD is the rhombus.

2. Square

All sides equal, all angles 90°.

When side is given:

1. Draw side AB.
2. Construct 90° at A and B.
3. Mark points D and C at distance = side.
4. Join CD.

When diagonal is given: Same as rhombus construction using diagonals (because diagonals of square are equal and bisect at 90°).

3. Rectangle

Opposite sides equal, all angles 90°.

When two adjacent sides given:

1. Draw AB (length).
2. Construct 90° at A and B, mark AD and BC (breadth).
3. Join DC.

When diagonals and one side given: More complicated, rarely asked in Class 8.

4. Parallelogram

Opposite sides equal and parallel.

Common construction: Three sides and one angle (included).

But most common is using two sides and included angle.

Important Points to Remember

• A quadrilateral has 4 sides, so total degrees of freedom = 4 × 2 = 8, but rigidity condition requires 5 measurements for unique construction.
• If only 4 measurements are given → infinite quadrilaterals possible.
• Rough sketch before actual construction helps avoid mistakes.
• Always label points clearly.
• Use sharp pencil and accurate compass setting.
• While constructing angles, protractor is allowed in Class 8.
• In exams, steps of construction must be written clearly with proper justification.

Common Mistakes

• Forgetting to draw rough figure.
• Incorrect compass radius setting.
• Not drawing arcs on correct side.
• Joining wrong points.
• Measuring angle on wrong side (reflex instead of given angle).

Summary Table of Constructions

Download pdf notes and Sulutions of the Chapter:

Chapter 5: Data Handling

 

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