Class 8 Maths Chapter 4 Note - Practical Geometry

Introduction

A quadrilateral is a polygon that has 2 diagonals, 4 sides, and 4 vertices that make 4 angles. The sum of all the interior angles of a quadrilateral is 360°. Square, rectangle, rhombus, parallelogram and kite are special types of quadrilaterals.

A quadrilateral can be constructed when 5 of its measurements are given. We should first draw a rough sketch to get an idea of the construction steps. Let us see the different criteria.

Four Sides, One Diagonal

Let’s look at an example to understand the construction.
Construct a quadrilateral PQRS in which PQ = 6 cm, QR = 5 cm, RS = 4 cm, PS = 5.2 cm and diagonal PR = 7 cm.

Steps of Construction

● Draw PQ = 6 cm.
● Taking P as the centre and radius equal to 7 cm, draw an arc.
● Taking Q as the centre and radius equal to 5 cm, draw another arc intersecting the previous arc at R.
● Join QR and PR.
● Taking P as the centre, and radius equal to 5.2 cm, draw an arc.
● Taking R as the centre and radius equal to 4 cm, draw another arc intersecting the previous arc at S.
● Join PS and RS.

PQRS is the required quadrilateral.

Three Sides, Two Diagonals

We will understand the steps of construction with the help of an example.
Construct a quadrilateral ABCD in which AB = 5 cm, BC = 4 cm, AD = 3.8 cm, diagonal AC = 5.8 cm and diagonal BD = 6.1 cm.

Steps of Construction

● Draw AB = 5 cm.
● Taking A as the centre and radius equal to 5.8 cm, draw an arc.
● Taking B as the centre and radius equal to 4 cm, draw another arc intersecting the previous arc at C.
● Join BC and AC.
● Taking A as the centre, and radius equal to 3.8 cm, draw an arc.
● Taking B as the centre, and radius equal to 6.1 cm draw another arc, intersecting the previous arc at D.
● Join AD and CD.

ABCD is the required quadrilateral.



Want to download NCERT solutions for class 8 maths Practical Geometrys? Get access to free PDF below:

NCERT Solutions for Class 8 Maths Chapter 4
NCERT Solutions Class 8 Maths Chapter 4
NCERT Solutions Class 8 Maths Exercise 4.1
NCERT Solutions Class 8 Maths Exercise 4.2
NCERT Solutions Class 8 Maths Exercise 4.3
NCERT Solutions Class 8 Maths Exercise 4.4
NCERT Solutions Class 8 Maths Exercise 4.5

Three Sides, Two Included Angles

Construct a quadrilateral MNOL such that MN = 6.2 cm, NO = 8 cm, LM = 5 cm, ∠M = 120° and ∠N = 90°.

Steps of Construction

● Draw the side MN of length 6.2 cm.
● At point M, draw an angle of 120°.
● At point N, draw an angle of 90°.
● By taking 8 cm as the radius and N as the centre, draw an arc and mark the point as O on the line of 90°.
● By taking 5 cm as the radius and M as the centre, draw an arc and mark the point as L on the line of 120°.
● Join points L and O.

MNOL is the required quadrilateral.

Three Angles, Two Adjacent Sides

Construct a quadrilateral ABCD such that AB = 5 cm, BC = 4.9 cm, ∠A = 75°, ∠B = 120°, ∠C = 90°.

Steps of Construction

● Draw side AB of length 5 cm.
● At point A, we draw an angle of 75°.
● At point B, we draw an angle of 120° and extend it till point X.
● By taking 4.9 cm as the radius and B as the centre, draw an arc and where the line OX and the arcs intersect, mark the point as C.
● At point C, we draw an angle of 90° and extend it till point Y.
● Mark point D where the lines from point A and C intersect each other.

ABCD is the required quadrilateral.

Let's move to the construction of the special quadrilaterals.

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Square

A square can be constructed if one of its sides is given.

Let us construct a square PQRS of side 4 cm.

In a square, all sides are equal, PQ = QR = RS = PS = 4 cm. All angles are right angles, ∠P = ∠Q = ∠R = ∠S = 90°.

Steps of Construction

● Draw side PQ of length 4 cm.
● At point P, we draw an angle of 90°.
● At point Q, we draw an angle of 90°.
● By taking 4 cm as the radius and P as the centre, draw an arc and mark the point as S.
● By taking 4 cm as the radius and Q as the centre, draw an arc and mark the point as R.
● Join the points S and R.

PQRS is the required square.

Let's move to the construction of rhombus.

Rhombus

A rhombus has four equal sides, opposite angles of equal measure and with diagonals bisecting each other at the right angles.

Let us construct a rhombus ABCD where diagonals BD = 5 cm and AC = 6 cm.

In a rhombus, the diagonals bisect each other at 90°. That is, diagonal BD is the perpendicular bisector of AC and vice-versa.

Let O be the midpoint of BD and AC. Therefore, OD = OB = 2.5 cm.

Steps of Construction

● Draw the diagonal AC = 6 cm.
● Draw perpendicular bisector of AC. With A as the centre, draw an arc on top and bottom of AC with a radius of more than half of the length of AC. With C as the centre and the same radius, draw an arc on top and bottom of AC intersecting the previous arcs.
● Join the points of arcs of the top and bottom of AC. Where this perpendicular bisector meets AC, mark it as point O.
● With O as the centre and radius of 2.5 cm, mark two arcs on perpendicular bisector opposite to each other. Label the points as B and D.
● Join AB, BC, CD, and DA.

ABCD is the required rhombus.

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