Geometrical Constructions (Exercise 19.6) RD Sharma Solutions | Mathematics (Maths) Class 6 PDF Download

 Page 1


 
 
 
 
 
 
                                                                           
1. Construct an angle of 60
o
 with the help of compasses and bisect it by paper folding. 
Solution: 
 
Construct a ray OA 
Taking O as centre and convenient radius, construct an arc which cuts the ray OA at the point P. 
Taking P as centre and same radius, construct another arc which cuts the previous arc at the point Q. 
Construct OQ and extend it to the point B. 
Here, ?AOB is the required angle of 60
o
 
Now cut the part of paper as sector OPQ 
Fold the part of paper such that the line segments OP and OQ coincides. 
The angle which is made at point O is the required angle which is half of ?AOB. 
 
 
2. Construct the following angles with the help of ruler and compasses only: 
(i) 30
o
 
(ii) 90
o
 
(iii) 45
o
 
(iv) 135
o
 
(v) 150
o
 
(vi) 105
o
 
Solution: 
 
(i) 30
o
 
Construct a ray OA. 
Taking O as centre and with convenient radius, construct an arc which cuts OA at the point P. 
Taking P as centre and same radius, construct an arc which cuts the previous arc at the point Q. 
Considering P and Q as centres and radius which is more than half of PQ construct two arcs which cuts each other 
and name it as point R. 
Construct OR and extend it to the point B. 
Here, ?AOB is the required angle of 30
o
 
Page 2


 
 
 
 
 
 
                                                                           
1. Construct an angle of 60
o
 with the help of compasses and bisect it by paper folding. 
Solution: 
 
Construct a ray OA 
Taking O as centre and convenient radius, construct an arc which cuts the ray OA at the point P. 
Taking P as centre and same radius, construct another arc which cuts the previous arc at the point Q. 
Construct OQ and extend it to the point B. 
Here, ?AOB is the required angle of 60
o
 
Now cut the part of paper as sector OPQ 
Fold the part of paper such that the line segments OP and OQ coincides. 
The angle which is made at point O is the required angle which is half of ?AOB. 
 
 
2. Construct the following angles with the help of ruler and compasses only: 
(i) 30
o
 
(ii) 90
o
 
(iii) 45
o
 
(iv) 135
o
 
(v) 150
o
 
(vi) 105
o
 
Solution: 
 
(i) 30
o
 
Construct a ray OA. 
Taking O as centre and with convenient radius, construct an arc which cuts OA at the point P. 
Taking P as centre and same radius, construct an arc which cuts the previous arc at the point Q. 
Considering P and Q as centres and radius which is more than half of PQ construct two arcs which cuts each other 
and name it as point R. 
Construct OR and extend it to the point B. 
Here, ?AOB is the required angle of 30
o
 
 
 
 
 
 
 
 
 
(ii) 90
o
 
Construct a ray OA. 
Taking O as centre and with convenient radius, construct an arc which cuts OA at the point P. 
Taking P as centre and same radius, construct an arc which cuts the previous arc at the point Q. 
Taking Q as centre and same radius, construct an arc which cuts the previous arc at the point R. 
Considering Q and R as centres and radius which is more than half of QR construct two arcs which cuts each 
other and name it as point S. 
Construct OS and extend it to the point B from the ray OB. 
Here, ?AOB is the required angle of 90
o
. 
 
 
(iii) 45
o
 
In order to draw an angle of 45
o
, construct an angle of 90
o
 and bisect it. 
Construct ?AOB = 90
o
 where OA and OB are the rays which intersect the arc at the points P and T 
Taking P and T as centres and radius which is more than half of PT, construct two arcs which cuts each other and 
name it as point X 
Construct OX and extend it to the point C to form the ray OC 
Here, ?AOC is the required angle of 45
o
. 
 
Page 3


 
 
 
 
 
 
                                                                           
1. Construct an angle of 60
o
 with the help of compasses and bisect it by paper folding. 
Solution: 
 
Construct a ray OA 
Taking O as centre and convenient radius, construct an arc which cuts the ray OA at the point P. 
Taking P as centre and same radius, construct another arc which cuts the previous arc at the point Q. 
Construct OQ and extend it to the point B. 
Here, ?AOB is the required angle of 60
o
 
Now cut the part of paper as sector OPQ 
Fold the part of paper such that the line segments OP and OQ coincides. 
The angle which is made at point O is the required angle which is half of ?AOB. 
 
 
2. Construct the following angles with the help of ruler and compasses only: 
(i) 30
o
 
(ii) 90
o
 
(iii) 45
o
 
(iv) 135
o
 
(v) 150
o
 
(vi) 105
o
 
Solution: 
 
(i) 30
o
 
Construct a ray OA. 
Taking O as centre and with convenient radius, construct an arc which cuts OA at the point P. 
Taking P as centre and same radius, construct an arc which cuts the previous arc at the point Q. 
Considering P and Q as centres and radius which is more than half of PQ construct two arcs which cuts each other 
and name it as point R. 
Construct OR and extend it to the point B. 
Here, ?AOB is the required angle of 30
o
 
 
 
 
 
 
 
 
 
(ii) 90
o
 
Construct a ray OA. 
Taking O as centre and with convenient radius, construct an arc which cuts OA at the point P. 
Taking P as centre and same radius, construct an arc which cuts the previous arc at the point Q. 
Taking Q as centre and same radius, construct an arc which cuts the previous arc at the point R. 
Considering Q and R as centres and radius which is more than half of QR construct two arcs which cuts each 
other and name it as point S. 
Construct OS and extend it to the point B from the ray OB. 
Here, ?AOB is the required angle of 90
o
. 
 
 
(iii) 45
o
 
In order to draw an angle of 45
o
, construct an angle of 90
o
 and bisect it. 
Construct ?AOB = 90
o
 where OA and OB are the rays which intersect the arc at the points P and T 
Taking P and T as centres and radius which is more than half of PT, construct two arcs which cuts each other and 
name it as point X 
Construct OX and extend it to the point C to form the ray OC 
Here, ?AOC is the required angle of 45
o
. 
 
 
 
 
 
 
 
(iv) 135
o
 
Construct a line AB and mark a point O in the middle of AB. 
Taking O as centre and convenient radius, construct an arc which cuts the line AB at the points P and Q. 
Construct an angle of 90
o
 on the ray OB as ?BOC = 90
o
 where OC cuts the arc at the point R 
Taking Q and R as centres and radius which is more than half of QR, construct two arcs which cuts each other and 
name it as point S. 
Construct OS and extend it to the point D to form the ray OD. 
Here, ?BOD is the required angle of 135
o
. 
 
 
(v) 150
o
 
Construct a line AB and mark a point O in the middle of AB. 
Taking O as centre and convenient radius construct an arc which cuts the line AB at the points P and Q. 
Taking Q as centre and same radius construct an arc which cuts the previous arc and name it as point R. 
Taking R as centre and same radius construct an arc which cuts the previous arc and name it as point S. 
Taking P and S as centres and radius which is more than half of PS, construct two arcs which cuts each other and 
name it as point T. 
Construct OT and extend it to the point C to form the ray OC. 
Here, ?BOC is the required angle of 150
o
. 
 
 
(vi) 105
o
 
Construct a ray OA and make ?AOB = 90
o
 and ?AOC = 120
o
 
Bisect ?BOC and get the ray OD. 
Here, ?AOD is the required angle of 105
o
. 
Page 4


 
 
 
 
 
 
                                                                           
1. Construct an angle of 60
o
 with the help of compasses and bisect it by paper folding. 
Solution: 
 
Construct a ray OA 
Taking O as centre and convenient radius, construct an arc which cuts the ray OA at the point P. 
Taking P as centre and same radius, construct another arc which cuts the previous arc at the point Q. 
Construct OQ and extend it to the point B. 
Here, ?AOB is the required angle of 60
o
 
Now cut the part of paper as sector OPQ 
Fold the part of paper such that the line segments OP and OQ coincides. 
The angle which is made at point O is the required angle which is half of ?AOB. 
 
 
2. Construct the following angles with the help of ruler and compasses only: 
(i) 30
o
 
(ii) 90
o
 
(iii) 45
o
 
(iv) 135
o
 
(v) 150
o
 
(vi) 105
o
 
Solution: 
 
(i) 30
o
 
Construct a ray OA. 
Taking O as centre and with convenient radius, construct an arc which cuts OA at the point P. 
Taking P as centre and same radius, construct an arc which cuts the previous arc at the point Q. 
Considering P and Q as centres and radius which is more than half of PQ construct two arcs which cuts each other 
and name it as point R. 
Construct OR and extend it to the point B. 
Here, ?AOB is the required angle of 30
o
 
 
 
 
 
 
 
 
 
(ii) 90
o
 
Construct a ray OA. 
Taking O as centre and with convenient radius, construct an arc which cuts OA at the point P. 
Taking P as centre and same radius, construct an arc which cuts the previous arc at the point Q. 
Taking Q as centre and same radius, construct an arc which cuts the previous arc at the point R. 
Considering Q and R as centres and radius which is more than half of QR construct two arcs which cuts each 
other and name it as point S. 
Construct OS and extend it to the point B from the ray OB. 
Here, ?AOB is the required angle of 90
o
. 
 
 
(iii) 45
o
 
In order to draw an angle of 45
o
, construct an angle of 90
o
 and bisect it. 
Construct ?AOB = 90
o
 where OA and OB are the rays which intersect the arc at the points P and T 
Taking P and T as centres and radius which is more than half of PT, construct two arcs which cuts each other and 
name it as point X 
Construct OX and extend it to the point C to form the ray OC 
Here, ?AOC is the required angle of 45
o
. 
 
 
 
 
 
 
 
(iv) 135
o
 
Construct a line AB and mark a point O in the middle of AB. 
Taking O as centre and convenient radius, construct an arc which cuts the line AB at the points P and Q. 
Construct an angle of 90
o
 on the ray OB as ?BOC = 90
o
 where OC cuts the arc at the point R 
Taking Q and R as centres and radius which is more than half of QR, construct two arcs which cuts each other and 
name it as point S. 
Construct OS and extend it to the point D to form the ray OD. 
Here, ?BOD is the required angle of 135
o
. 
 
 
(v) 150
o
 
Construct a line AB and mark a point O in the middle of AB. 
Taking O as centre and convenient radius construct an arc which cuts the line AB at the points P and Q. 
Taking Q as centre and same radius construct an arc which cuts the previous arc and name it as point R. 
Taking R as centre and same radius construct an arc which cuts the previous arc and name it as point S. 
Taking P and S as centres and radius which is more than half of PS, construct two arcs which cuts each other and 
name it as point T. 
Construct OT and extend it to the point C to form the ray OC. 
Here, ?BOC is the required angle of 150
o
. 
 
 
(vi) 105
o
 
Construct a ray OA and make ?AOB = 90
o
 and ?AOC = 120
o
 
Bisect ?BOC and get the ray OD. 
Here, ?AOD is the required angle of 105
o
. 
 
 
 
 
 
 
 
3. Construct a rectangle whose adjacent sides are 8 cm and 3 cm. 
Solution: 
 
Construct a line segment AB of length 8 cm. 
Draw ?BAX = 90
o
 at A and ?ABY = 90
o
 at B 
With the help of compass and ruler, mark a point D on the ray AX where AD = 3 cm 
In the same way mark the point C on the ray BY where BC = 3 cm 
Construct the line segment CD 
Hence, ABCD is the required rectangle.