4.2    CONSTRUCTING A QUADRILATERAL:

 1. If the four sides and a diagonal of the quadrilateral are given.

ExampleConstruct a quadrilateral ABCD in which AB   =   5 cm, BC   =   7 cm, CD   =   6 cm, DA = 6.5 cm  and  AC  =  8 cm.

Step 1:  ∆ ABC can be constructed using S.S.S criterion of the construction of triangle.

Step 2: Here  AC is the diagonal, so D will be somewhere opposite to B with reference to AC.

and  AD  = 6.5 cm so draw an arc from A as the centre with radius  6.5 cm.

Step 3: Now draw an arc with C as the centre and by taking radius 6 cm. This arc  should

intersect the above arc.

Step 4: The point of intersection of the two arcs is point D. Now join AD and DC to complete the quadrilateral.

Hence, ABCD is the required quadrilateral.

 

EXERCISE:   4.1

 Construct the following quadrilaterals:

(i)        Quadrilateral ABCD where  AB   =   4.5 cm, BC   =   5.5 cm,  CD   =   4 cm,

AD   =   6 cm,   AC   =   7 cm

(ii)       Quadrilateral JUMP where        JU   =  3.5 cm, UM   =   4 cm,  MP   =   5 cm,

PJ   =  4.5 cm,  PU   =   6.5 cm

(iii)     Parallelogram MORE where   OR   =   6 cm, RE   =   4.5 cm, EO   =   7.5 cm

(iv)      Rhombus BEST where   BE   =   4.5 cm, ET   =   6 cm

(i)        Given:  AB   = 4.5 cm,  BC  = 5.5 cm,  CD  =  4 cm,  AD  =  6 cm,  AC  =  7 cm

To construct: A quadrilateral ABCD

Steps of construction:

(a)   Draw AB   =   4.5 cm.

(b)   Draw an arc taking radius 5.5 cm from point B.

(c)   Taking radius 7 cm, draw an another arc from point A which intersects the first arc at point C.

(d)   Join BC and AC.

(e)   Draw an arc of radius 6 cm from point A and draw another arc of radius 4 cm from point C which intersects at D.

(f)   Join AD and CD.

NCERT Solution For Class 8 Maths Chapter 4 Image 4

The required quadrilateral  is  ABCD.

(ii) Given:  JU  =  3.5 cm,   UM  =  4 cm,  MP  =  5 cm,  PJ  =  4.5 cm,  PU  =  6.5 cm

To construct: A quadrilateral JUMP

Steps of construction:

(a)  Draw JU   =   3.5 cm.

(b)  Draw an arc of radius 4.5 cm taking centre J and then draw another arc of radius 6.5 cm taking U as centre. Both arcs intersect at P.

(c)  Join PJ and PU.

(d) Draw arc of radius 5 cm and 4 cm taking P and U as centres respectively, which intersect at M.

(e)  Join Mp and MU.

NCERT Solution For Class 8 Maths Chapter 4 Image 8

The  required quadrilateral is  JUMP.

(iii) Given:  OR  =  6 cm,  RE  =  4.5 cm,   EO  =  7.5 cm

To construct: A parallelogram MORE.

Steps of construction:

(a)   Draw OR   =   6 cm.

(b)   Draw arcs of radius 7.5 cm and radius 4.5 cm taking O and R as centres respectively, which

intersect at E.

(c ) Join OE and RE.

(d)   Draw an arc of 6 cm radius taking E as center.

(e)   Draw another arc of 4.5 cm radius taking O as center, which intersects at M.

(f)   Join OM and EM.

NCERT Solution For Class 8 Maths Chapter 4 Image 12

The  required parallelogram  is MORE.

(iv)   Given:  BE  =  4.5 cm,  ET  =  6 cm

To construct: A rhombus BEST.

Steps of construction:

(a)   Draw TE   =   6 cm and bisect it into two equal parts.

(b)   Draw up and down perpendiculars to TE.

(c)   Draw two arcs of  4.5 cm taking E and T as centers, which intersect at S.

(d)   Again draw two arcs of 4.5 cm taking E and T as centers, which intersects at B.

(e)   Join TS, ES, BT and EB.

NCERT Solution For Class 8 Maths Chapter 4 Image 16

The required rhombus  is BEST.

Additional Questions:

  1. Construct a quadrilateral BEST, given ES =   4.5cm, BT  =   5.5 cm, St  =   5 cm, the diagonal BS  =   5.5 cm and diagonal ET  =   7 cm. Find Angle E, Angle T and RE.
  2. Draw a line segment of length 10 cm and divide it into 4 equal parts.
  3. Construct a quadrilateral PQRS with PQ =   4 cm, QR   =   3 cm, PS   =   2.5 cm, PR   =   4.5 cm and QS   =   4 cm

 

 If  two diagonals and three sides of the quadrilateral are given:

Example:   Construct a quadrilateral ABCD if the two diagonals are AC =  6.5 cm and BD  =  8 cm. The other sides are BC  =  5.5 cm,  AD  = 6.5 cm and   CD  =  6 cm.

First of all, draw a rough sketch of the quadrilateral by using the given measurements. Then start constructing the real one.

Step 1: We can see that AD, AC and DC are given so we can construct a triangle Δ ACD by using S.S.S criterion.

Step 2: Now, BD is given so we can draw the point B keeping D as the centre and draw an arc of radius 8 cm just opposite to the point D with reference to AC.

Step 3: BC is given so we can draw an arc keeping C as centre and radius 5.5 cm so that it intersects the other arc.

Step 4: That point of intersection of the arcs is point B. Join AB and BC to complete the quadrilateral.

Thus  ABCD is the required quadrilateral.

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

 (i) Given:  LI  =  4 cm, IF  =  3 cm, TL  =  2.5 cm, LF  =  4.5 cm, IT  =  4 cm

To construct: A quadrilateral LIFT

Steps of construction:

(a)   Draw a line segment LI   =   4 cm.

(b)   Taking radius 4.5 cm, draw an arc taking L as center.

(c)   Draw an arc of 3 cm taking I as center which intersects the first arc at F.

(d)   Join FI and FL.

(e)   Draw another arc of radius 2.5 cm taking L as center and 4 cm taking I as center which

intersect at T.

(f) Join TF, Tl and TI.

NCERT Solution For Class 8 Maths Chapter 4 Image 20

The required quadrilateral   is LIFT.

(ii) Given:  OL   =   7.5 cm, GL   =   6 cm, GD   =   6 cm, LD   =   5 cm, OD   =   10 cm

To construct: A quadrilateral GOLD

Steps of construction:

(a)   Draw a line segment OL   =   7.5 cm

(b)   Draw an arc of radius 5 cm taking L as center and another arc of radius 10 cm taking O as

center which intersect the first arc point at D.

(c)   Join LD and OD.

(d)   Draw an arc of radius 6 cm from D and draw another arc of radius 6 cm taking L as center,

which intersects at G.

(e)   Join GD and GO.

NCERT Solution For Class 8 Maths Chapter 4 Image 24

The required quadrilateral  is GOLD.

(iii) Given:  BN   =   5.6 cm, DE   =   6.5 cm

To construct: A rhombus BEND.

Steps of construction:

(a)   Draw DE   =   6.5 cm.

(b)   Draw perpendicular bisector of line segment DE.

(c)   Draw two arcs of radius 2.8 cm from intersection point O, which intersects the line

KN at B   and N.

(d)   Join BE,  BD as well as  ND and NE.

NCERT Solution For Class 8 Maths Chapter 4 Image 28

The required rhombus   is BEND.

Additional Questions:

  1. Construct a quadrilateral ABCD,  given that BC =   4.5 cm,   AD  =  5.5 cm,  CD  =  5 cm the diagonal AC  =  5.5 cm and diagonal  BD  =  7 cm.
  2. Construct a quadrilateral DEFG,  given that EF =  5.5 cm,  DG  =  6.5 cm,  FG  =  5 cm the diagonal  DF  =  6.5 cm and diagonal EG  =  6.7 cm

 

  1. If three angles and two adjacent sides of the quadrilateral are given.

Example: Construct a quadrilateral ABCD in which the two adjacent sides are

AB  =  4.5 cm and BC  =  7.5 cm. The given three angles are  ∠A  =  75ᵒ,  ∠B  =  105ᵒ and  ∠ C   =   120ᵒ.

Solution: Draw a rough sketch so that construction becomes easy.

Step 1: Draw AB  =  4.5 cm. Then measure  ∠B  =  105° using protractor and draw BC  =  7.5 cm.

Step 2: Draw  ∠ C  =  120°.

Step 3: Measure  ∠ A  =  75° and make a line until it touches the line coming from point C.

So ABCD is the required quadrilateral.

EXERCISE 4.3

  1. Construct the following quadrilaterals:

(i)        Quadrilateral MORE

MO   =   6 cm,   OR   =   4.5 cm,     ∠  M   =   60     ∠ O   =   105     ∠  R   =   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

 (i) Given:  MO   =   6 cm,   OR   =   4.5 cm,    ∠ M   =   60    ∠ O   =   105     ∠ R   =   105

 To construct: A quadrilateral MORE.

Steps of construction:

(a)   Draw a line segment MO  =  6 cm.

(b)   Construct  ∠O  =   105 and taking the  radius as 4.5 cm,  taking O as  centre, draw an arc  which   intersects at R.

(c)   Also construct an angle 105 at R and produce the side RE.

(d)   Construct another angle of 60 at point M and the side ME. Both  the sides ME and RE   intersect at E.

NCERT Solution For Class 8 Maths Chapter 4 Image 32

The required quadrilateral is MORE.

(ii)       Given:  PL   =   4 cm, LA   =   6.5 cm,     ∠ P   =   90 ∠ A   =   110 ∠ N   =   85

To construct: A quadrilateral PLAN.

To find:   ∠ L   =   360 – (90 +110   + 85 )

=   360   –  285  =  75

Steps of construction:

(a)   Draw a line segment PL   =   4 cm.

(b)   Construct angle of 90 at P and produce the side PN.

(c)   Construct angle of 75 at L and with L as center, draw an arc of radius 6 cm, which

intersects at A.

(d)   Construct  ∠ A   =   110 at A and produce the side AN which intersects PN at N.

NCERT Solution For Class 8 Maths Chapter 4 Image 36

The required quadrilateral  is PLAN.

(iii)     Given:  HE  =  5 cm, EA  =  6 cm,     ∠ R  =  85

To construct: A parallelogram HEAR.

To find:   ∠ H   =   180   –  85   = 95      ( Sum of adjacent angles of a parallelogram is 180)

Steps of construction:

(a)   Draw a line segment HE   =   5 cm.

(b)   Construct  ∠ H   =   95 and draw an arc of radius 6 cm with center H. It intersects AR at R.

(c)   Join RH.

(d)   Draw  ∠ R  =   ∠ E  =  85  and draw an arc of radius 6 cm with E as a centre which

intersects RA at A.

(e)   Join RA

The required parallelogram  is HEAR.

( iv)     Given:  OK   =   7 cm, KA   =   5 cm

To construct: A rectangle OKAY.

Steps of Construction:

(a)   Draw a line segment OK   =   7 cm.

(b)   Construct angle 90 at both points O and K and produce these sides.

(c)   Draw two arcs of radius 5 cm from points O and K respectively.

These arcs intersect at Y and A.

(d)   Join YA.

NCERT Solution For Class 8 Maths Chapter 4 Image 44

The required rectangle  is OKAY.

ADDITIONAL QUESTIONS:

  1. Construct a quadrilateral ABCD in which AB =  5 cm, BC  =  6.5 cm, angle A  =   5°, angle B  =   105° and angle C  =  120°.
  2. Construct a quadrilateral PQRS in which PQ =  7 cm, QR  =  7.5 cm, angle P  =  75°, angle Q  =  105° and angle R  =  120°.

 

 If the three sides with two included angles of the quadrilateral are given.

Example:Construct a quadrilateral ABCD in which the three sides are AB  =  5 cm,

BC  =  6 cm and CD  = 7.5 cm. The two included angles are   B  =  105°  and   C   =   80°.

 Solution:   Draw a rough sketch.

Step 1: Draw the line BC   =   6 cm. Then draw   ∠ B   =   105° and mark the length of AB   =   5 cm.

Step 2: Draw   ∠ C   =   80° using protractor towards point B.

Step 3: Mark the length of CD i.e.7.5 cm from C to make CD   =   7.5 cm.

Step 4: Join AD which will complete the quadrilateral ABCD.

Hence ABCD is the required quadrilateral.

EXERCISE 4.4

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 cm,      R   =   75 ∠ U   =   120

(i) Given:  DE   =   4 cm, EA   =   5 cm,  AR   =   4.5 cm,  ∠ E   =   60 ∠ A   =   90

To construct: A quadrilateral DEAR.

Steps of construction:

(a)   Draw a line segment DE   =   4 cm.

(b)   At point E, construct an angle of 60

(c)   Taking radius 5 cm, draw an arc from point E which intersects at A.

(d)   Construct  ∠ A   =   90 draw an arc of radius 4.5 cm with center A which intersect at R.

(e)   Join RD.

NCERT Solution For Class 8 Maths Chapter 4 Image 48

The required quadrilateral  is  DEAR.

(ii)       Given:  TR   =   3.5 cm,  RU   =   3 cm,   UE   =   4 cm,  ∠ R   = 75   ∠ U   =   120

To construct: A quadrilateral TRUE

Steps of construction:

(a)   Draw a line segment TR   =   3.5 cm.

(b)   Construct an angle 75 at R and draw an arc of radius 3 cm with R as center,

which intersects at U.

(c)   Construct an angle of 120 at U and produce the side UE.

(d)   Draw an arc of radius 4 cm with U as center.

(e)   Join UE and TE.

NCERT Solution For Class 8 Maths Chapter 4 Image 52

The required quadrilateral  is TRUE.

Additional Questions

  1. Construct a quadrilateral ABCD, where AB =   4 cm, BC   =   5 cm, CD   =   6.5 cm and ∟B   =   105° and ∟C   =  80°.
  2. Construct a quadrilateral PQRS, where PQ =   5 cm, QR   =   5 cm, RS   =   7.5 cm and ∟Q   =   115° and ∟R   =   85°.