NCERT solutions for class 8 Maths Chapter 4 Practical Geometry

Solution for Exercise 4.1

1. Construct the following quadrilaterals:

Construct the following quadrilaterals.

(i) Quadrilateral ABCD

AB = 4.5 cm, BC = 5.5 cm, CD = 4 cm, AD = 6 cm, AC = 7 cm

(ii) Quadrilateral JUMP

JU = 3.5 cm, UM = 4 cm, MP = 5 cm, PJ = 4.5 cm, PU = 6.5 cm

(iii) Parallelogram MORE

OR = 6 cm, RE = 4.5 cm, EO = 7.5 cm

(iv) Rhombus BEST

BE = 4.5 cm, ET = 6 cm


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Answer :

(i) Draw a rough sketch first as per the given dimensions.


Construction:
Step 1: Draw a line segment AB = 4.5 cm

Step 2: Draw an arc with center as B and of radius 5.5 cm.

Step 3: Similarly, draw another arc with center as A and radius 7 cm to meet the previous arc at C.

Step 4: Now taking radius 4 cm, draw an arc with centre C and also draw another arc with center A and radius 6 cm to cut the former arc at D.

Step 5: Join BC, AC, CD and AD.

Hence, we have got the required figure as per the given data.

(ii) Draw a rough sketch first as per the given dimensions.



Construction:
Step 1: Draw a line segment JU = 3.5 cm.

Step 2: Draw an arc with centre at J and radius 4.5 cm.

Step 3: Draw another arc with centre at U and radius 6.5 cm to meet the previous arc at P and join JP and UP.

Step 4: Now with radius 4 cm, draw an arc with centre at U.

Step 5: Draw another arc with centre P and radius 5 cm to meet the previous arc at M. Step 6: Join UM and PM.

Hence we have, JUMP as the required quadrilateral.
(iii) Draw a rough sketch first as per the given dimensions and we know that opposite sides of a parallelogram are equal.



Construction:
Step 1: Draw a line segment OR = 6 cm.
Step 2: Draw an arc with centre at R and of radius 4.5 cm.

Step 3: Draw another arc with centre at O and radius 7.5 cm to meet the previous arc at E and join RE and OE.

Step 4: Draw an arc with centre at E and radius 6 cm.

Step 5: Draw another arc with centre at O and radius 4.5 cm to meet the former arc at M. Step 6: Join EM and OM.

Hence we have, MORE as the required parallelogram with given dimensions.

(iv) Draw a rough sketch first as per the given dimensions and we know that all sides of a rhombus are equal.



Construction:

Step 1: Draw BE = 4.5 cm

Step 2: Draw an arc with centre B and radius 4.5 cm.

Step 3: Draw another arc with centre E and radius 6 cm to meet the previous arc at T and join BT and ET.

Step 4: Draw two arcs with centres E and T with equal radii 4.5 cm to meet each other at S and join ES and TS, and join ES and TS.

Hence we have, BEST as the required rhombus with given dimensions.

Solution for Exercise 4.2

1.Construct the following quadrilaterals.
(i) Quadrilateral LIFT

LI = 4 cm

IF = 3 cm

TL = 2.5 cm

LF = 4.5 cm

IT = 4 cm

(ii) Quadrilateral GOLD

OL = 7.5 cm

GL = 6 cm

GD = 6 cm

LD = 5 cm

OD = 10 cm

(iii) Rhombus BEND

BN = 5.6 cm

DE = 6.5 cm


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Answer :

(i)Draw a rough sketch first as per the given dimensions.

Construction:

Step 1: Draw a line segment LI = 4 cm.

Step 2: Draw an arc with centre at I and radius 3 cm.

Step 3: Draw another arc with centre L and radius 4.5 cm to meet the former arc at F and Join LF and IF.

Step 4: Draw an arc with centre L and radius 2.5 cm.

Step 5: Draw another arc with centre I and radius 4 cm to meet the previous arc at T and Join LT and IT.

Hence we have LIFT as the required quadrilateral with given dimensions.

(ii)Draw a rough sketch first as per the given dimensions.


Construction:

Step 1: Draw a line segment OL = 7.5 cm

Step 2: Draw an arc with centre at O and of radius 10 cm.

Step 3: Draw another arc with centre at L and of radius 5 cm to meet the previous arc at D and join OD and LD.

Step 4: Draw an arc with centre L and D with equal radii of 6 cm to meet each other at G and join LG and DG.

Hence we have GOLD as the required quadrilateral with given dimensions.

(iii)Draw a rough sketch first as per the given dimensions and we know that the diagonals of a rhombus bisect each other at the right angle.


Construction:

Step 1: Draw a line segment BN = 5.6 cm.

Step 2: Draw the right bisector of BN at O.

Step 3: Draw two arcs with centre O and radius \(\frac { 1 }{ 2 } \times DE\), i.e., \(\frac { 1 }{ 2 } \times 6.5 = 3.25 cm\) to meet the right bisector at D and E.

Step 4: Join BE, EN, ND and BD.

Hence we have, BEND as the required rhombus with given dimensions.

Solution for Exercise 4.3

1.Construct the following quadrilaterals:

(i) Quadrilateral MORE

MO = 6 cm ∠R = \(105^{\circ}\), OR = 4.5 cm, ∠M = \(60^{\circ}\), ∠O = \(105^{\circ}\)

(ii) Quadrilateral PLAN

PL = 4 cm, LA = 6.5 cm, ∠P = \(90^{\circ}\), ∠A = \(110^{\circ}\), ∠N = \(85^{\circ}\)

(iii) Parallelogram HEAR

HE = 5 cm, EA = 6 cm, ∠R = \(85^{\circ}\)

(iv) Rectangle OKAY

OK = 7 cm, KA = 5 cm


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Answer :

(i) Draw a rough sketch first as per the given dimensions.


Construction:

Step 1: Draw a line segment OR = 4.5 cm

Step 2: Draw an angle of \(105^{\circ}\) at O and also an angle of \(105^{\circ}\) at R with the help of protractor.

Step 3: Make an arc OM = 6 cm.

Step 4: Draw an angle of \(60^{\circ}\) at M to meet the line through R at E.

Hence we have, MORE as the required quadrilateral.

(ii)Draw a rough sketch first as per the given dimensions.


Construction:

Step 1: Draw a line segment LA = 6.5 cm

Step 2: Draw an angle of 75° at L and \(110^{\circ}\) at A with the help of a protractor. [\(∵ 360^{\circ} – (110^{\circ} + 90^{\circ} + 85^{\circ}) = 75^{\circ}\)]

Step 3: Cut LP = 4 cm.

Step 4: Draw an angle of 90° at P which meets the line through A at N.

Hence we have PLAN as the required quadrilateral.

(iii)Draw a rough sketch first as per the given dimensions and we know that opposite sides of a parallelogram are equal.


Construction:

Step 1: Draw a line segment HE = 5 cm.

Step 2: Draw an angle of \(85^{\circ}\) at E and cut an arc of EA = 6 cm.

Step 3: Draw an arc with centre A and radius 5 cm.

Step 4: Draw another arc with centre H and radius 6 cm to meet the previous arc at R and join HR and AR.

Hence we have, HEAR as the required parallelogram.

(iv)Draw a rough sketch first as per the given dimensions and we know that each angle of a rectangle is \(90^{\circ}\) and opposite sides are equal.


Construction:

Step 1: Draw a line segment OK = 7 cm.

Step 2: Draw the angle of \(90^{\circ}\) at K and cut an arc of KA = 5 cm.

Step 3: Draw an arc with centre at O and radius 5 cm.

Step 4: Draw another arc with centre at A and radius 7 cm to meet the previous arc at Y and join OY and AY.

Hence we have OKAY as the required rectangle.

Solution for Exercise 4.4

1.Construct the following quadrilaterals:
(i) Quadrilateral DEAR

DE = 4 cm, EA = 5 cm, AR = 4.5 cm, ∠E = \(60^{\circ}\), ∠A =\( 90^{\circ}\)

(ii) Quadrilateral TRUE

TR = 3.5 cm, RU = 3 cm, UE = 4.5 cm, ∠R = \(75^{\circ}\), ∠U =\( 120^{\circ}\)


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Answer :

(i)Draw a rough sketch first as per the given dimensions


Construction:

Step 1: Draw a line segment DE = 4 cm.

Step 2: Draw an angle of \(60^{\circ}\) at E.

Step 3: Draw an arc with centre E and radius 5 cm to meet the angle line at A.

Step 4: Draw an angle of \(90^{\circ}\) at A and cut an arc AR = 4.5 cm and join DR.

Hence we have, DEAR as the required quadrilateral.

(ii)Draw a rough sketch first as per the given dimensions


Construction:

Step 1: Draw a line segment TR = 3.5 cm

Step 2: Draw an angle of \(75^{\circ}\) at R and cut RU = 3 cm.

Step 3: Draw an angle of \(120^{\circ}\) at U and cut UE = 4.5 cm.

Step 4: Join TE.

Hence we have TRUE as the required quadrilateral.

Solution for Exercise 4.5

1.The square READ with RE = 5.1 cm.


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Answer :

Construction:



Step 1: Draw RE = 5.1 cm.

Step 2: Draw an angle of \(90^{\circ}\) at E and cut an arc EA = 5.1 cm.

Step 3: Draw two arcs from A and R with radius 5.1 cm to cut each other at D.

Step 4: Join RD and AD.

Hence we have, READ as the required square.

2.A rhombus whose diagonals are 5.2 cm and 6.4 cm long.


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Answer :

Construction:


Step 1: Draw a line segment AC = 6.4 cm.

Step 2: Draw the right angle bisector of AC at E.

Step 3: Draw two arcs with centre at E and radius = \(\frac { 5.2 }{ 2 }\) = 2.6 cm to cut the previous diagonal at B and D.

Step 4: Join AD, AB, BC and DC.

Hence we have ABCD as the required rhombus.

3.A rectangle with adjacent sides of lengths 5 cm and 4 cm.


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Answer :

Construction:


Step 1: Draw PQ = 5 cm.

Step 2: Draw an angle of \(90^{\circ}\) at Q and cut an arc QR = 4 cm.

Step 3: Draw an arc with centre R and radius 5 cm.

Step 4: Draw another arc with centre P and radius 4 cm to meet the previous arc at S.

Step 5: Join RS and PS.

Hence we have PQRS as the required rectangle.

4.A parallelogram OKAY where OK = 5.5 cm and KA = 4.2 cm. Is it unique?


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Answer :

Construction:


Step 1: Draw a line segment OK = 5.5 cm.

Step 2: Draw an angle of any measure (let's say \(60^{\circ}\)) at K and cut an arc KA = 4.2 cm.

Step 3: Draw an arc with centre A and radius of 5.5 cm.

Step 4: Draw another arc with centre O and radius 4.2 cm to cut the previous arc at Y.

Step 5: Join AY and OY.

Hence we have OKAY as the required parallelogram.

No, it is not a unique parallelogram because the angle at K can be of any measure other than \(60^{\circ}\).



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