ML Aggarwal Solution Class 9 Chapter 13 Rectilinear Figures Exercise 13.2

 Exercise 13.2

Question 1

Using ruler and compasses only, construct the quadrilateral ABCD in which ∠ BAD = 45°, AD = AB = 6cm, BC = 3.6cm, CD = 5cm. Measure ∠ BCD.

Sol :
Steps of construction :
(i) Draw a line segment AB=6 cm












(ii) At A, draw a ray AX making an angle of 45° and cut off AD=6 cm

(iii) With centre B and radius 3.6 cm and with centre D and radius 5 cm , draw two arcs intersecting each other at C

(iv) Join BC and DC
ABCD is the required quadrilateral
On measuring ∠BCD , it is 60°

Question 2

Draw a quadrilateral ABCD with AB = 6cm, BC = 4cm, CD = 4 cm and ∠ ABC = ∠ BCD = 90°

Sol :
Steps of construction :
(i) Draw a line segment BC=4 cm
(ii) At B and C draw rays BX and CY making an angle of 90° each















(iii) From BX, cut off BA=6 cm and from CY, cut off CD=4 cm

(iv) Join AD,

ABCD is the required quadrilateral 


Question 3

Using ruler and compasses only, construct the quadrilateral ABCD given that AB = 5 cm, BC = 2.5 cm, CD = 6 cm, ∠BAD = 90° and the diagonal AC = 5.5 cm.

Sol :
Steps of construction :
(i) Draw a line segment AB=5 cm
(ii) With centre A and radius 5.5 cm and with centre B and radius 2.5 cm draw arcs which intersects each other at C
(iii) Join AC and BC









(iv) At A , draw a ray AX making an angle of 90°

(v) With centre C and radius 6 cm, draw an arc intersecting AX and D

(vi) Join CD

ABCD is the required quadrilateral


Question 4

Construct a quadrilateral ABCD in which AB = 3.3 cm, BC = 4.9 cm, CD = 5.8 cm, DA = 4 cm and BD = 5.3 cm.

Sol :
Steps of construction :
(i) Draw a line segment AB=3.3 cm
(ii) With centre A and radius 4 cm, and which centre B and radius 5.3 cm , draw arcs intersecting each other at D













(iii) Join AD and BD

(iv) With centre B and radius 4.9 cm and with centre D and radius 5.8 cm , draw arcs intersecting each other at C

(v) Join BC and DC

ABCD is the required quadrilateral


Question 5

Construct a trapezium ABCD in which AD || BC, AB = CD = 3 cm, BC = 5.2cm and AD = 4 cm

Sol :
Steps of construction :
(i) Draw a line segment BC=5.2 cm
(ii) From BC, cut off BE=AD=4 cm
(iii) With centre E and C, and radius 3 cm draw arcs intersecting each other at D












(iv) Join ED and CD

(v) With centre D and radius 4 cm and with centre B and radius 3 cm, draw arcs intersecting each other at A.

(vi) Join BC and DA

ABCD is the required trapezium


Question 6

Construct a trapezium ABCD in which AD || BC, ∠B= 60°, AB = 5 cm. BC = 6.2 cm and CD = 4.8 cm.

Sol ::
Steps of construction :
(i) Draw a line segment BC=6.2 cm
(ii) At B, draw a ray BX making an angle of 60° and cut off AB=5 cm
(iii) From A, draw a line AY parallel to BC











(iv) With centre C and radius 4.8 cm, draw an arc which intersects AY at D and D' 

(v) Join CD and CD'

Then ABCD and ABCD' are the required two trapezium


Question 7

Using ruler and compasses only, construct a parallelogram ABCD with AB = 5.1 cm, BC = 7 cm and ∠ABC = 75°.

Sol :
Steps of construction :
(i) Draw a line segment BC=7 cm
(ii) A to B, draw a ray Bx making an angle of 75° and cut off AB=5.1 cm 
(iii) With centre A and radius 7 cm with centre C and radius 5.1 cm, draw arcs intersecting each other at D
(iv) Join AD and CD
ABCD is the required parallelogram










Question 8

Using ruler and compasses only, construct a parallelogram ABCD in which AB = 4.6 cm, BC = 3.2 cm and AC = 6.1 cm.

Sol :









(i) Draw a line segment AB=4.6 cm

(ii) With centre A and radius 6.1 cm and with centre B and radius 3.2 cm, draw arcs intersecting each other at C

(iii) Join AC and BC

(iv) Again with centre A and radius 3.2 cm and with centre C and radius 4.6 cm , draw arcs intersecting each other at D

(v) Join AD and CD

Then ABCD is the required parallelogram 


Question 9

Using ruler and compasses, construct a parallelogram ABCD give that AB = 4 cm, AC = 10 cm, BD = 6 cm. Measure BC.

Sol :
Given : AB=4 cm, AC=10 cm, BD=6 cm
Required : 
(i) To construct a parallelogram ABCD
(ii) Length of BC










Steps of construction :

(1) Construct triangle OAB such that

⇒$\mathrm{OA}=\frac{1}{2} \times \mathrm{AC}=\frac{1}{2} \times 10 \mathrm{~cm}$=5 cm

⇒$\mathrm{OB}=\frac{1}{2} \times \mathrm{BD}=\frac{1}{2} \times 6 \mathrm{~cm}$=3 cm

(Since diagonals of parallelogram bisect each other) and AB=4 cm


(2) Produce AO, to C such that OA=OC=5 cm

(3) Produce BO, to D such that OB=OD=3 cm

(4) Join AD , BC and CD

(5) ABCD is the required parallelogram

(6) Measure BC which is equal to 7.2 cm


Question 10

Using ruler and compasses only, construct a parallelogram ABCD such that BC = 4 cm, diagonal AC = 8.6 cm and diagonal BD = 4.4 cm. Measure the side AB.

Sol :
Given : BC=4 cm , Diagonal AC=8.6 cm and diagonal BP=4.4 cm
Required : (i) To construct a parallelogram 
(ii) Measure the side AB










Steps of Construction :
(1) Construct triangle OBC such that 

$\mathrm{OB}=\frac{1}{2} \times \mathrm{BD}=\frac{1}{2} \times 4.4 \mathrm{~cm}$=2.2 cm

$\mathrm{OC}=\frac{1}{2} \times \mathrm{AC}=\frac{1}{2} \times 8.6 \mathrm{~cm}$=4.3 cm

(2) Produce BO to D such tha BO=OD=2.2 cm

(3) Produce CO to A such that CO=OA=4.3 cm

(4) Join AB, AD and CD

(5) ABCD is the required parallelogram

(6) Measure the side AB, AB=5.6 cm


Question 11

Use ruler and compasses to construct a parallelogram with diagonals 6 cm and 8 cm in length having given the acute angle between them is 60°. Measure one of the longer sides.

Sol :
Given : Diagonals AC=6 cm , Diagonal BD=8 cm
Angle between the diagonals=60°

Required : 

(i) To construct a parallelogram

(ii) To measure one of longer side







Steps of construction :

(1) Draw AC=6 cm

(2) Find the mid point O of AC

(∴Diagonals of parallelogram bisect each other)

(3) Draw line POQ such that ∠POC=60° and OB=OD$=\frac{1}{2} \mathrm{BD}=\frac{1}{2} \times 8 \mathrm{~cm}$=4 cm

∴From OP cut OD=4 cm and from OQ cut OB=4 cm

(4) Join AB, BC, CD and DA

(5) ABCD is the required parallelogram

(6) Measure the length of side AD=6.1 cm


Question 12

Using ruler and compasses only, draw a parallelogram whose diagonals are 4 cm and 6 cm long and contain an angle of 75°. Measure and write down the length of one of the shorter sides of the parallelogram.

Sol :
Steps of construction :
(i) Draw a line segment AC=6 cm
(ii) Bisect AC at O
(iii) At O, draw a ray XY making an angle of 75° at O
(iv) From OX and OY , cut off OD=OB$=\frac{4}{2}$=2 cm



















(v) Join AB, BC , CD and DA
Then ABCD is the required parallelogram
On measuring one of the shorter sides
AB=CD=3 cm


Question 13

Using ruler and compasses only, construct a parallelogram ABCD with AB = 6 cm, altitude = 3.5 cm and side BC = 4 cm. Measure the acute angles of the parallelogram.
Sol :
Given : AB=6 cm, Altitude=3.5 cm and BC=4 cm
Required : 
(i) To construct a parallelogram ABCD
(ii) To measure the acute angle of parallelogram 








 
Steps of construction :

(1) Draw AB=6 cm

(2) At B, draw BP⟂AB

(3) From BP, cut BE=3.5 cm=height of parallelogram

(4) Through E draw QR parallel to AB

(5) With B as centre and radius BC=4 cm draw an arc which cuts QR at C

(6) Since opposite sides of parallelogram are equal 

∴AD=BC=4 cm

∴With A as centre and radius=4 cm draw an arc which cut QR at D

(7) ∴ABCD is the required parallelogram

(8) To measure the acute angle of parallelogram which is equal to 61


Question 14

The perpendicular distances between the pairs of opposite sides of a parallelogram ABCD are 3 cm and 4 cm and one of its angles measures 60°. Using ruler and compasses only, construct ABCD.

Sol :
Given : ∠BAD=60°
Height be 3 cm and 4 cm from AB and BC respectively (say)

Required : To construct a parallelogram ABCD






Steps of construction :

(1) Draw a straight line PQ, take a point A on it 

(2) At A , construct ∠QAF=60°

(3) At A, draw AE⊥PQ from AE cut off AN=3 cm

(4) Through N draw a straight line parallel to PQ to meet AF at D

(5) At A, draw AG⊥AD, from AG cut off AM=4 cm

(6) Through M, draw a straight line parallel to AD to meet AQ in B and ND in C. Then ABCD is the required parallelogram.


Question 15

Using ruler and compasses, construct a rectangle ABCD with AB = 5cm and AD = 3 cm.

Sol :
Steps of construction :
(1) Draw a straight line AB=5 cm
(2) At A and B construct ∠XAB and ∠YBA=90°
(3) From A and B cut off AC and BD=3 cm each 
(4) Join CD
(5) ABCD is the required rectangle













Question 16

Using ruler and compasses only, construct a rectangle each of whose diagonals measures 6cm and the diagonals intersect at an angle of 45°.
Sol :
Steps of construction :
(1) Draw a line segment AC=6 cm
(2) Bisect it at O
(3) At O , draw a ray XY making an angle of 45° at O
(4) From XY, cut off
OB=OD$=\frac{6}{2}=3$ cm each
(5) Join AB, BC, CD and DA
Then ABCD is the required rectangle


Question 17

Using ruler and compasses only, construct a square having a diagonal of length 5cm. Measure its sides correct to the nearest millimeter.

Sol :
Steps of construction :
(1) Draw a line segment AC=5 cm
(2) Draw its perpendicular bisector XY bisecting it at O


















(3) From XY , cut off

OB=OD$=\frac{5}{2}=2.5$ cm

(4) Join AB, BC, CD and DA

ABCD is the required square

On measuring its sides each side=3.6 cm (approximately)


Question 18

Using ruler and compasses only construct A rhombus ABCD given that AB 5cm, AC = 6cm measure ∠BAD.

Sol :
(i) Draw a line segment AB=5 cm













(ii) With centre A and radius 6 cm, with centre B and radius 5 cm, draw arcs intersecting each other at C

(iii) Join AC and BC

(iv) With centre A and C and radius 5 cm, draw arcs intersecting each otther at D

(v) Join AD and CD

Then ABCD is a rhombus 

On measuring ∠BAD=106°


Question 19

Using ruler and compasses only, construct rhombus ABCD with sides of length 4cm and diagonal AC of length 5 cm. Measure ∠ABC.

Sol :
Steps of construction :
(i) Draw a line segment AC=5 cm
(ii) With centre A and C and radius 4 cm draw arcs intersecting each other above and below AC at D and B
(iii) Join AB, BC , CD and DA
ABCD is the required rhombus










Question 20

Construct a rhombus PQRS whose diagonals PR and QS are 8cip and 6cm respectively.

Sol :
Steps of construction :
(i) Draw a line segment PR=8 cm
(ii) Draw its perpendicular bisector XY intersecting it at O
(iv) From XY, cut off OQ=OS
$=\frac{6}{2}$=3 cm each










(iv) Join PQ, QR, RS and SP

Then PQRS is the required rhombus


Question 21

Construct a rhombus ABCD of side 4.6 cm and ∠BCD = 135°, by using ruler and compasses only.

Sol :
Steps of construction :
(i) Draw a line segment BC=4.6 cm
(ii) AT C, draw a ray CX making an angle of 135° and cut off CD=4.6 cm










(iii) With centres B and D, and radius 4.6 cm draw arcs intersecting each other at A.

(iv) Join BA, DA

Then ABCD is the required rhombus


Question 22

Construct a trapezium in which AB || CD, AB = 4.6 cm, ∠ ABC = 90°, ∠ DAB = 120° and the distance between parallel sides is 2.9 cm.

Sol :
Steps of construction :
(i) Draw a line segment AB=4.6 cm
(ii) At B, draw a ray BZ making an angle of 90° and cut off BC=2.9 cm (distance between AB and CD)






(iii) At C, draw a parallel line XY to AB.

(iv) At A, draw a ray making an angle of 120° meeting XY at D

Then ABCD is the required trapezium


Question 23

Construct a trapezium ABCD when one of parallel sides AB = 4.8 cm, height = 2.6cm, BC = 3.1 cm and AD = 3.6 cm.

Sol :
Steps of construction :
(1) Draw a line segment AB=4.8 cm













(2) At A draw a ray AZ making an angle of 90° and cut off AL=2.6 cm
(3) At L, draw a line XY parallel to AB
(4) With centre A and radius 3.1 cm , draw arcs intersecting XY at D and C respectively.
(5) Join AD , BC
Then ABCD is the required trapezium

Question 24

Construct a regular hexagon of side 2.5 cm.
Sol :
Given : Each side of regular hexagon = 2.5 cm
Required : To construct a regular hexagon











Steps of construction :
(1) With O as centre and radius=2.5 cm , draw a circle
(2) Take any point A on the circumference of circle.
(3) With A as centre and radius equal to 2.5 cm, draw an arc which cuts the circumference in B.
(4) With B as centre and radius =2.5 cm, draw an arc which circumference of circle at C.
(5) With C as centre and radius =2.5 cm draw an arc which cuts circumference of circle at $D$.
(6) With D as centre and radius =2.5 cm draw an arc which cuts circumference of circle at E
(7) With E as centre and radius =2.5 cm draw an arc which cuts circumference of circle at F
(8) Join AB,BC,CD,DE,EF and FA.
(9) ABCDEF is the required Hexagon.

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