CLASS 8TH CHAPTER 4- PRACTICAL GEOMERTY NCERT SOLUTION

Exercise 4.1 | Class 8th Mathematics

1. 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
   Solution:
   (i) We have to draw first rough sketch.
   
   Construction:
   Step I: Draw AB = 4.5 cm
   Step II: Draw an arc with centre B and radius 5.5 cm.
   Step III: Draw another arc with centre A and radius 7 cm to meet the previous arc at C.
   Step IV: Draw an arc with centre C and radius 4 cm.
   Step V: Draw another arc with centre A and radius 6 cm to cut the former arc at D.
   Step VI: Join BC, AC, CD and AD.
   (ii) We have to draw the first rough sketch.
   
   Thus ABCD is the required quadrilateral.
   Construction:
   Step I: Draw JU = 3.5 cm.
   Step II: Draw an arc with centre J and radius 4.5 cm.
   Step III: Draw another arc with centre U and radius 6.5 cm to meet the previous arc at P.
   Step IV: Join JP and UP.
   Step V: Draw an arc with centre U and radius 4 cm.
   Step VI: Draw another arc with centre P and radius 5 cm to meet the previous arc at M.
   Step VII: Join UM and PM.
   Thus, JUMP is the required quadrilateral.
   (iii) We have to draw the first rough sketch.
   
   Construction: (Opposite sides of a parallelogram are equal)
   Step I: Draw OR = 6 cm.
   Step II: Draw an arc with centre R and radius 4.5 cm.
   Step III: Draw another arc with centre O and radius 7.5 cm to meet the previous arc at E.
   Step IV: Join RE and OE.
   Step V: Draw an arc with centre E and radius 6 cm.
   Step VI: Draw another arc with centre O and radius 4.5 cm to meet the former arc at M.
   Step VII: Join EM and OM.
   Thus, MORE is the required parallelogram.
   (iv) We have to draw first rough sketch.
   
   Construction: (All sides of a rhombus are equal)
   Step I: Draw BE = 4.5 cm
   Step II: Draw an arc with centre B and radius 4.5 cm.
   Step III: Draw another arc with centre E and radius 6 cm to meet the previous arc at T.
   Step IV: Join BT and ET.
   Step V: Draw two arcs with centres E and T with equal radii 4-5 cm to meet each other at S. .
   Step VI: Join ES and TS.
   Thus, BEST is the required rhombus.

Exercise 4.2 | Class 8th Mathematics

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
   Solution:
   (i) Construction:
   
   Step I: Draw LI = 4 cm.
   Step II: Draw an arc with centre I and radius 3 cm.
   Step III: Draw another arc with centre L and radius 4.5 cm to meet the former arc at F.
   Step IV: Join LF and IF.
   Step V: Draw an arc with centre L and radius 2.5 cm.
   Step VI: Draw another arc with centre I and radius 4 cm to meet the previous arc at T.
   Step VII: Join LT and IT.
   Thus LIFT is the required quadrilateral.
   (ii) Construction:
   
   Step I: Draw OL = 7.5 cm
   Step II: Draw an arc with centre O and radius 10 cm.
   Step III: Draw another arc with centre L and radius 5 cm to meet the previous arc at D.
   Step IV: Join OD and LD.
   Step V: Draw an arc with centre L and D with equal radii of 6 cm to meet each other at G.
   Step VI: Join LG and DG.
   Thus GOLD is the required quadrilateral.
   (iii) Construction: (The diagonals of a rhombus bisect each other at the right angle)
   
   Step I: Draw BN = 5.6 cm.
   Step II: Draw the right bisector of BN at O.
   Step III: Draw two arcs with centre O and radius 12 × DE, i.e., 12 × 6.5 = 3.25 cm to meet the right bisector at D and E.
   Step IV: Join BE, EN, ND and BD.
   Thus, BEND is the required rhombus.

Exercise 4.3 | Class 8th Mathematics

1. Construct the following quadrilaterals:
   (i) Quadrilateral MORE
   MO = 6 cm, ∠R = 105°, OR = 4.5 cm, ∠M = 60°, ∠O = 105°
   (ii) Quadrilateral PLAN
   PL = 4 cm, LA = 6.5 cm, ∠P = 90°, ∠A = 110°, ∠N = 85°
   (iii) Parallelogram HEAR
   HE = 5 cm, EA = 6 cm, ∠R = 85°
   (iv) Rectangle OKAY
   OK = 7 cm, KA = 5 cm
   Solution:
   (i) Construction:
   Step I: Draw OR = 4.5 cm
   Step II: Draw two angles of 105° each at O and R with the help of protactor.
   Step III: Cut OM = 6 cm.
   
   Step IV: Draw an angle of 60° at M to meet the angle line through R at E.
   Thus, MORE is the required quadrilateral.

(ii) Construction:

Step I: Draw LA = 6.5 cm
Step II: Draw an angle of 75° at L and 110° at A with the help of a protractor.
[∵ 360° – (110° + 90° + 85°) = 75°]
Step III: Cut LP = 4 cm.
Step IV: Draw an angle of 90° at P which meets the angle line through A at N.
Thus PLAN is the required quadrilateral.

(iii) Construction: (Opposite sides of a parallelogram are equal)

Step I: Draw HE = 5 cm.
Step II: Draw an angle of 85° at E and cut EA = 6 cm.
Step III: Draw an arc with centre A and radius 5 cm.
Step IV: Draw another arc with centre H and radius 6 cm to meet the previous arc at R.
Step V: Join HR and AR
Thus, HEAR is the required parallelogram.

(iv) Construction:
(Each angle of a rectangle is 90° and opposite sides are equal.)

Step I: Draw OK = 7 cm.
Step II: Draw the angle of 90° at K and cut KA = 5 cm.
Step III: Draw an arc with centre O and radius 5 cm.
Step IV: Draw another arc with centre A and radius 7 cm to meet the previous arc at Y.
Step V: Join OY and AY.
Thus OKAY is the required rectangle.

Exercise 4.3 | Class 8th Mathematics

1. Construct the following quadrilaterals:
   (i) Quadrilateral DEAR
   DE = 4 cm, EA = 5 cm, AR = 4.5 cm, ∠E = 60°, ∠A = 90°
   (ii) Quadrilateral TRUE
   TR = 3.5 cm, RU = 3 cm, UE = 4.5 cm, ∠R = 75°, ∠U = 120°
   Solution:
   (i) Construction:
   
   Step I: Draw DE = 4 cm.
   Step II: Draw an angle of 60° at E.
   Step III: Draw an arc with centre E and radius 5 cm to meet the angle line at A.
   Step IV: Draw an angle of 90° at A and cut AR = 4.5 cm.
   Step V: Join DR.
   Thus, DEAR is the required quadrilateral.

(ii) Construction:

Step I: Draw TR = 3.5 cm
Step II: Draw an angle of 75° at R and cut RU = 3 cm.
Step III: Draw an angle of 120° at U and cut UE = 4.5 cm.
Step IV: Join TE.
Thus, TRUE is the required quadrilateral.

Exercise 4.5 | Class 8th Mathematics

1. The square READ with RE = 5.1 cm.
   Solution:
   Construction:
   
   Step I: Draw RE = 5.1 cm.
   Step II: Draw an angle of 90° at E and cut EA = 5.1 cm.
   Step III: Draw two arcs from A and R with radius 5.1 cm to cut each other at D.
   Step IV: Join RD and AD.
   Thus, READ is the required square.

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

Step I: Draw AC = 6.4 cm.
Step II: Draw the right bisector of AC at E.
Step III: Draw two arcs with centre E and radius = 5.22 = 2.6 cm to cut the previous diagonal at B and D.
Step IV: Join AD, AB, BC and DC.
Thus ABCD is the required rhombus.

3. A rectangle with adjacent sides of lengths 5 cm and 4 cm.
Solution:
Construction: Let the two adjacent sides of a rectangle PQRS be PQ = 5 cm and QR = 4 cm.

Step I: Draw PQ = 5 cm.
Step II: Draw an angle of 90° at Q and cut QR = 4 cm.
Step III: Draw an arc with centre R and radius 5 cm.
Step IV: Draw another arc with centre P and radius 4 cm to meet the previous arc at S.
Step V: Join RS and PS.
Thus, PQRS is the required rectangle.

4. A parallelogram OKAY where OK = 5.5 cm and KA = 4.2 cm. Is it unique?
Solution:
Construction:
Step I: Draw OK = 5.5 cm.
Step II: Draw an angle of any measure (say 60°) at K and cut KA = 4.2 cm.

Step III: Draw an arc with centre A and radius of 5.5 cm.
Step IV: Draw another arc with centre O and radius 4.2 cm to cut the previous arc at Y.
Step V: Join AY and OY.
Thus, OKAY is the required parallelogram.
No, it is not a unique parallelogram. The angle at K can be of measure other than 60°.

Extra Questions | Class 8th Mathematics

Question 1.
Construct a quadrilateral PQRS, given that QR = 4.5 cm, PS = 5.5 cm, RS = 5 cm and the diagonal PR = 5.5 cm and diagonal SQ = 7 cm.

Solution:
Construction:
Step I: Draw QR = 4.5 cm.
Step II: Draw an arc with centre R and radius 5 cm.
Step III: Draw another arc with centre Q and radius 7 cm to meet the previous arc at S.
Step IV: Join RS and QS.
Step V: Draw two arcs with centre S and R and radius 5.5 cm each to meet each other at P.
Step VI: Join RP, SP and PQ.
Thus PQRS is the required quadrilateral.

2. Construct a quadrilateral ABCD in which AB = 4 cm, BC = 3.5 cm, CD = 5 cm, AD = 5.5 cm and ∠B = 75°.

Solution:
Construction:
Step I: Draw AB = 4 cm.
Step II: Draw an angle of 75° at B and cut BC = 3.5 cm.
Step III: Draw an arc with centre C and radius 5 cm.
Step IV: Draw another arc with centre A and radius 5.5 cm to meet the previous arc at D.
Step V: Join CD and AD.
Thus ABCD is the required quadrilateral.

3. Construct a square whose side is 5 cm.

Solution:
Construction:
Step I: Draw AB = 5 cm.
Step II: Draw an angle of 90° at B and cut BC = 5 cm.
Step III: Draw two arcs with centre A and C and same radii of 5 cm which meet each other at D.
Step IV: Join AD and CD.
Thus, ABCD is the required square.

4. Construct a rhombus ABCD in which AB = 5.8 cm and AC = 7.5 cm.

Solution:
Construction:
Step I: Draw AB = 5.8 cm.
Step II: Draw an arc with centre B and radius 5.8 cm.
Step III: Draw another arc with centre A and radius 7.5 cm to meet the previous arc at C.
Step IV: Draw two arcs with centres A and C and of the same radius 5.8 cm to meet each other at D.
Step V: Join BC, AC, CD and AD.
Thus ABCD is the required rhombus.

5. Construct a rhombus whose diagonals are 6 cm and 8 cm.

Solution:
Construction:
Step I: Draw SQ = 8 cm.
Step II: Draw a right bisector of SQ at O.
Step III: Draw two arcs with centre O and radius 3 cm each to cut the right bisector at P and R.
Step TV: Join PQ, QR, RS and SP.
Thus PQRS is the required rhombus.

6. Construct a rectangle whose diagonal is 5 cm and the angle between the diagonal is 50°.


Solution:
Construction:
Step I: Draw AC = 5 cm.
Step II: Draw the right bisector of AC at O.
Step III: Draw an angle of 50° at O and product both sides.
Step IV: Draw two arcs with centre O and of the same radius 2.5 cm to cut at B and D.
Step V: Join AB, BC, CD and DA.
Thus, ABCD is the required rectangle.

7. Construct a quadrilateral ABCD in which BC = 4 cm, ∠B = 60°, ∠C = 135°, AB = 5 cm and ∠A = 90°.

Solution:
Construction:
Step I: Draw AB = 5 cm.
Step II: Draw the angle of 60° at B and cut BC = 4 cm.
Step III: Draw an angle of 135° at C and angle of 90° at A which meet each other at D.
Thus, ABCD is the required quadrilateral.

8. Construct a parallelogram ABCD in which AB = 5.5 cm, AC = 7 cm and BD = 8 cm.

Solution:
Construction:
Step I: Draw AB = 5.5 cm.
Step II: Draw an arc with centre B and radius 82 cm = 4 cm.
Step III: Draw another arc with centre A and radius 72 cm = 3.5 cm which cuts the previous arc at O.
Step IV: Join AO and produce to C such that AO = OC.
Step V: Join BO and produce to D such that BO = OD.
Step VI: Join BC, CD and AD.
Thus ABCD is the required parallelogram.

9. Construct a rhombus PAIR, given that PA = 6 cm and angle ∠A = 110°.
Solution:
Since in a rhombus, all sides are equal, so PA = AI = IR = RP = 6 cm
Also, rhombus is a parallelogram
so, adjacent angle, ∠I = 180° – 110° = 70°

Steps of construction
Step I. Draw AI = 6 cm
Step II. Draw ray AX¯ such that ∠IAX = 110° and draw IY¯ such that ∠AIY = 70°.
Step III. With A and I as centres and radius 6 cm draw arcs intersecting AX and IY at P and R respectively.
Step IV. Join PR.
Thus, PAIR is the required rhombus.