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Textbook: Geometrical Constructions | Mathematics Class 6 (Maharashtra Board) PDF Download

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 Page 1


87
Can you tell?
(1) When constructing a building, what is the method
used to make sure that a wall is exactly upright?
What does the mason in the picture have in his
hand?
What do you think is his purpose for using it?
(2) Have you looked at lamp posts on the roadside?
   The Perpendicular
 In the figure here, line l and line n intersect at point M. 
Measure every angle formed at the point M. 
  If an angle between line l and line n is a right angle, we 
say that the lines are perpendicular to each other. This is 
written as ‘line l ? line n’ in symbols. It is read as ‘Line l 
is perpendicular to line n’. 
?? 	
Drawing a perpendicular to a line at a point on the line.
(1) Using a set square
?? Draw line PQ. Take point R anywhere
on the line. 
?? Place the set square on the line in such 
a way that the vertex of its right angle 
is at point R and one arm of the right 
angle falls on line PQ.
?? Draw a line RS along the other arm of 
the right angle of the set square. 
?? The line RS is perpendicular to the line 
PQ at R.
17 Geometrical Constructions
How do they stand?
Try this.
P
R
Q
Q
S
P
R
Page 2


87
Can you tell?
(1) When constructing a building, what is the method
used to make sure that a wall is exactly upright?
What does the mason in the picture have in his
hand?
What do you think is his purpose for using it?
(2) Have you looked at lamp posts on the roadside?
   The Perpendicular
 In the figure here, line l and line n intersect at point M. 
Measure every angle formed at the point M. 
  If an angle between line l and line n is a right angle, we 
say that the lines are perpendicular to each other. This is 
written as ‘line l ? line n’ in symbols. It is read as ‘Line l 
is perpendicular to line n’. 
?? 	
Drawing a perpendicular to a line at a point on the line.
(1) Using a set square
?? Draw line PQ. Take point R anywhere
on the line. 
?? Place the set square on the line in such 
a way that the vertex of its right angle 
is at point R and one arm of the right 
angle falls on line PQ.
?? Draw a line RS along the other arm of 
the right angle of the set square. 
?? The line RS is perpendicular to the line 
PQ at R.
17 Geometrical Constructions
How do they stand?
Try this.
P
R
Q
Q
S
P
R
88
(2) Using a protractor
 ?? Draw line RS. Take point M anywhere 
on the line.
 ?? 	In order to draw a perpendicular 
through M, place the centre of the 
protractor on point M, as shown.
 ?? Mark a point N at the 90° mark on 
the protractor.  
	 ?? Draw a line passing through points M 
and N.
 ?? The line MN is perpendicular to line 
RS at M.
  Line MN ? line RS.
(3) Using a compass
  
 
		 	 ?? Draw line MN. Take point K anywhere on the 
line.
		 	 ?? Place the compass point on point K. Draw two 
arcs on either side of point K to cut the line 
MN at equal distances from K. Name the points 
of intersection A and B respectively.
		 	 ?? Place the compass point at A and, taking a 
convenient distance greater than half the length 
of AB, draw an arc on one side of the line.
		 	 ?? Place the compass point at B and using the 
same distance, draw another arc to intersect 
the first one at T.
	 		 ?? Draw a line passing through points K and T.
   ?? 	 The line KT is perpendicular to line MN at K. 
  
  Line KT ? line MN.
 
 Why must we take a distance greater than half of the length of AB? 
What will happen if we take a smaller distance?
S
S
T
M
A K
B N
Think about it.
R M
R
M
N
M
A
K B N
Page 3


87
Can you tell?
(1) When constructing a building, what is the method
used to make sure that a wall is exactly upright?
What does the mason in the picture have in his
hand?
What do you think is his purpose for using it?
(2) Have you looked at lamp posts on the roadside?
   The Perpendicular
 In the figure here, line l and line n intersect at point M. 
Measure every angle formed at the point M. 
  If an angle between line l and line n is a right angle, we 
say that the lines are perpendicular to each other. This is 
written as ‘line l ? line n’ in symbols. It is read as ‘Line l 
is perpendicular to line n’. 
?? 	
Drawing a perpendicular to a line at a point on the line.
(1) Using a set square
?? Draw line PQ. Take point R anywhere
on the line. 
?? Place the set square on the line in such 
a way that the vertex of its right angle 
is at point R and one arm of the right 
angle falls on line PQ.
?? Draw a line RS along the other arm of 
the right angle of the set square. 
?? The line RS is perpendicular to the line 
PQ at R.
17 Geometrical Constructions
How do they stand?
Try this.
P
R
Q
Q
S
P
R
88
(2) Using a protractor
 ?? Draw line RS. Take point M anywhere 
on the line.
 ?? 	In order to draw a perpendicular 
through M, place the centre of the 
protractor on point M, as shown.
 ?? Mark a point N at the 90° mark on 
the protractor.  
	 ?? Draw a line passing through points M 
and N.
 ?? The line MN is perpendicular to line 
RS at M.
  Line MN ? line RS.
(3) Using a compass
  
 
		 	 ?? Draw line MN. Take point K anywhere on the 
line.
		 	 ?? Place the compass point on point K. Draw two 
arcs on either side of point K to cut the line 
MN at equal distances from K. Name the points 
of intersection A and B respectively.
		 	 ?? Place the compass point at A and, taking a 
convenient distance greater than half the length 
of AB, draw an arc on one side of the line.
		 	 ?? Place the compass point at B and using the 
same distance, draw another arc to intersect 
the first one at T.
	 		 ?? Draw a line passing through points K and T.
   ?? 	 The line KT is perpendicular to line MN at K. 
  
  Line KT ? line MN.
 
 Why must we take a distance greater than half of the length of AB? 
What will happen if we take a smaller distance?
S
S
T
M
A K
B N
Think about it.
R M
R
M
N
M
A
K B N
89
1. Draw line l. Take any point P on the line. Using a set square, draw a line perpendicular 
to line l at the point P.  
2. Draw a line AB. Using a compass, draw a line perpendicular to AB at the point B.  
3. Draw line CD. Take any point M on the line. Using a protractor, draw a line 
perpendicular to line CD at the point M. 
?? 	 Drawing a perpendicular to a line from a point outside the line.
 (1) By folding the paper
 ?? Draw a line MN on a paper. 
  Take a point P anywhere outside the line.
 ?? Keeping the line MN in view, fold the paper along 
the line MN.
 ?? Now fold the paper through point P in such a way 
that the part of line MN on one side of the fold 
falls on the part of line MN on the other side 
of the fold.  
 ?? 	 Unfold the paper. Name the point of intersection 
of the two folds Q. Draw the line PQ. This line 
falls on a fold in the paper.
  Using a protractor, measure every angle formed 
at the point Q .
  Line PQ is perpendicular to line MN.
  Line 
 
PQ
 ? line  
MN.
 (2) Using a set square    
  ?? 	 Draw line XY. Take point P anywhere outside XY.
  ?? Place one of the arms of the right angle of a set 
square along the line XY.
Practice Set 39
M N
P
M N
P
M
P
Q
M
N
Q
P
Page 4


87
Can you tell?
(1) When constructing a building, what is the method
used to make sure that a wall is exactly upright?
What does the mason in the picture have in his
hand?
What do you think is his purpose for using it?
(2) Have you looked at lamp posts on the roadside?
   The Perpendicular
 In the figure here, line l and line n intersect at point M. 
Measure every angle formed at the point M. 
  If an angle between line l and line n is a right angle, we 
say that the lines are perpendicular to each other. This is 
written as ‘line l ? line n’ in symbols. It is read as ‘Line l 
is perpendicular to line n’. 
?? 	
Drawing a perpendicular to a line at a point on the line.
(1) Using a set square
?? Draw line PQ. Take point R anywhere
on the line. 
?? Place the set square on the line in such 
a way that the vertex of its right angle 
is at point R and one arm of the right 
angle falls on line PQ.
?? Draw a line RS along the other arm of 
the right angle of the set square. 
?? The line RS is perpendicular to the line 
PQ at R.
17 Geometrical Constructions
How do they stand?
Try this.
P
R
Q
Q
S
P
R
88
(2) Using a protractor
 ?? Draw line RS. Take point M anywhere 
on the line.
 ?? 	In order to draw a perpendicular 
through M, place the centre of the 
protractor on point M, as shown.
 ?? Mark a point N at the 90° mark on 
the protractor.  
	 ?? Draw a line passing through points M 
and N.
 ?? The line MN is perpendicular to line 
RS at M.
  Line MN ? line RS.
(3) Using a compass
  
 
		 	 ?? Draw line MN. Take point K anywhere on the 
line.
		 	 ?? Place the compass point on point K. Draw two 
arcs on either side of point K to cut the line 
MN at equal distances from K. Name the points 
of intersection A and B respectively.
		 	 ?? Place the compass point at A and, taking a 
convenient distance greater than half the length 
of AB, draw an arc on one side of the line.
		 	 ?? Place the compass point at B and using the 
same distance, draw another arc to intersect 
the first one at T.
	 		 ?? Draw a line passing through points K and T.
   ?? 	 The line KT is perpendicular to line MN at K. 
  
  Line KT ? line MN.
 
 Why must we take a distance greater than half of the length of AB? 
What will happen if we take a smaller distance?
S
S
T
M
A K
B N
Think about it.
R M
R
M
N
M
A
K B N
89
1. Draw line l. Take any point P on the line. Using a set square, draw a line perpendicular 
to line l at the point P.  
2. Draw a line AB. Using a compass, draw a line perpendicular to AB at the point B.  
3. Draw line CD. Take any point M on the line. Using a protractor, draw a line 
perpendicular to line CD at the point M. 
?? 	 Drawing a perpendicular to a line from a point outside the line.
 (1) By folding the paper
 ?? Draw a line MN on a paper. 
  Take a point P anywhere outside the line.
 ?? Keeping the line MN in view, fold the paper along 
the line MN.
 ?? Now fold the paper through point P in such a way 
that the part of line MN on one side of the fold 
falls on the part of line MN on the other side 
of the fold.  
 ?? 	 Unfold the paper. Name the point of intersection 
of the two folds Q. Draw the line PQ. This line 
falls on a fold in the paper.
  Using a protractor, measure every angle formed 
at the point Q .
  Line PQ is perpendicular to line MN.
  Line 
 
PQ
 ? line  
MN.
 (2) Using a set square    
  ?? 	 Draw line XY. Take point P anywhere outside XY.
  ?? Place one of the arms of the right angle of a set 
square along the line XY.
Practice Set 39
M N
P
M N
P
M
P
Q
M
N
Q
P
90
 ??  Slide the set square along the line in such a way 
that the other arm of its right angle touches point 
P. Draw a line along this side, passing through 
point P. Name the line PS.
   Measure the angles to verify that the line is a 
perpendicular.
(3) Using a compass and ruler
   
	 ??  Draw line MN. Take any point K outside the line.
	 ??  Placing the compass point at point K and using 
any convenient distance, draw arcs to cut the line 
MN at two points A and B.
	 ??  Place the compass point at A and taking a 
distance greater than half of AB, draw an arc on 
the lower side of line MN.
	 ??  Place the compass point at B and using the            
same distance, draw an arc to cut the previous 
arc at T.
	 ??  Draw the line KT. 
	 ??  Line KT is perpendicular to line MN. Verify. 
   
 
The Perpendicular Bisector
 A wooden ‘yoke’ is used for pulling a 
bullock cart. 
 How is the position of the yoke determined? 
 To do that, a rope is used to measure equal 
distances from the spine/ midline of the bullock 
cart. Which geometrical property is used here?
 Find out from the craftsmen or from other 
experienced persons, why this is done.
Think about it.
M
N
K
T
M
N
K
T
A B
In the above construction, why must the distance in the compass be kept constant?
Page 5


87
Can you tell?
(1) When constructing a building, what is the method
used to make sure that a wall is exactly upright?
What does the mason in the picture have in his
hand?
What do you think is his purpose for using it?
(2) Have you looked at lamp posts on the roadside?
   The Perpendicular
 In the figure here, line l and line n intersect at point M. 
Measure every angle formed at the point M. 
  If an angle between line l and line n is a right angle, we 
say that the lines are perpendicular to each other. This is 
written as ‘line l ? line n’ in symbols. It is read as ‘Line l 
is perpendicular to line n’. 
?? 	
Drawing a perpendicular to a line at a point on the line.
(1) Using a set square
?? Draw line PQ. Take point R anywhere
on the line. 
?? Place the set square on the line in such 
a way that the vertex of its right angle 
is at point R and one arm of the right 
angle falls on line PQ.
?? Draw a line RS along the other arm of 
the right angle of the set square. 
?? The line RS is perpendicular to the line 
PQ at R.
17 Geometrical Constructions
How do they stand?
Try this.
P
R
Q
Q
S
P
R
88
(2) Using a protractor
 ?? Draw line RS. Take point M anywhere 
on the line.
 ?? 	In order to draw a perpendicular 
through M, place the centre of the 
protractor on point M, as shown.
 ?? Mark a point N at the 90° mark on 
the protractor.  
	 ?? Draw a line passing through points M 
and N.
 ?? The line MN is perpendicular to line 
RS at M.
  Line MN ? line RS.
(3) Using a compass
  
 
		 	 ?? Draw line MN. Take point K anywhere on the 
line.
		 	 ?? Place the compass point on point K. Draw two 
arcs on either side of point K to cut the line 
MN at equal distances from K. Name the points 
of intersection A and B respectively.
		 	 ?? Place the compass point at A and, taking a 
convenient distance greater than half the length 
of AB, draw an arc on one side of the line.
		 	 ?? Place the compass point at B and using the 
same distance, draw another arc to intersect 
the first one at T.
	 		 ?? Draw a line passing through points K and T.
   ?? 	 The line KT is perpendicular to line MN at K. 
  
  Line KT ? line MN.
 
 Why must we take a distance greater than half of the length of AB? 
What will happen if we take a smaller distance?
S
S
T
M
A K
B N
Think about it.
R M
R
M
N
M
A
K B N
89
1. Draw line l. Take any point P on the line. Using a set square, draw a line perpendicular 
to line l at the point P.  
2. Draw a line AB. Using a compass, draw a line perpendicular to AB at the point B.  
3. Draw line CD. Take any point M on the line. Using a protractor, draw a line 
perpendicular to line CD at the point M. 
?? 	 Drawing a perpendicular to a line from a point outside the line.
 (1) By folding the paper
 ?? Draw a line MN on a paper. 
  Take a point P anywhere outside the line.
 ?? Keeping the line MN in view, fold the paper along 
the line MN.
 ?? Now fold the paper through point P in such a way 
that the part of line MN on one side of the fold 
falls on the part of line MN on the other side 
of the fold.  
 ?? 	 Unfold the paper. Name the point of intersection 
of the two folds Q. Draw the line PQ. This line 
falls on a fold in the paper.
  Using a protractor, measure every angle formed 
at the point Q .
  Line PQ is perpendicular to line MN.
  Line 
 
PQ
 ? line  
MN.
 (2) Using a set square    
  ?? 	 Draw line XY. Take point P anywhere outside XY.
  ?? Place one of the arms of the right angle of a set 
square along the line XY.
Practice Set 39
M N
P
M N
P
M
P
Q
M
N
Q
P
90
 ??  Slide the set square along the line in such a way 
that the other arm of its right angle touches point 
P. Draw a line along this side, passing through 
point P. Name the line PS.
   Measure the angles to verify that the line is a 
perpendicular.
(3) Using a compass and ruler
   
	 ??  Draw line MN. Take any point K outside the line.
	 ??  Placing the compass point at point K and using 
any convenient distance, draw arcs to cut the line 
MN at two points A and B.
	 ??  Place the compass point at A and taking a 
distance greater than half of AB, draw an arc on 
the lower side of line MN.
	 ??  Place the compass point at B and using the            
same distance, draw an arc to cut the previous 
arc at T.
	 ??  Draw the line KT. 
	 ??  Line KT is perpendicular to line MN. Verify. 
   
 
The Perpendicular Bisector
 A wooden ‘yoke’ is used for pulling a 
bullock cart. 
 How is the position of the yoke determined? 
 To do that, a rope is used to measure equal 
distances from the spine/ midline of the bullock 
cart. Which geometrical property is used here?
 Find out from the craftsmen or from other 
experienced persons, why this is done.
Think about it.
M
N
K
T
M
N
K
T
A B
In the above construction, why must the distance in the compass be kept constant?
91
The Perpendicular Bisector of a Line Segment
Line p and line q pass through the point M on seg AB.
Line p and line q are bisectors of the segment AB.
Measure the angle between line p and seg AB.
Of the two lines p and q, line p is a bisector and also 
perpendicular to seg AB. 
Hence, line p is called the perpendicular bisector of
seg AB.
Why is line q not a perpendicular bisector of seg AB?
?? 	 Drawing the perpendicular bisector of a segment, using a compass.
 ?? Draw seg AB. 
 ?? Place the compass point at A
and taking a distance greater 
than half the length of seg AB, 
draw two arcs, one below and
one above seg AB. 
 ?? Place the compass point at B and 
using  the  same distance draw
arcs to intersect the previous 
arcs at P and Q. Draw line PQ.
?? The line PQ is the perpendicular bisector of seg AB. Verify.
 
 
Activity :  Take a rectangular sheet of paper. Fold the paper so that the lower edge 
of the paper falls on its top edge and fold it over again from right to left. 
Observe the two folds that have formed on the paper. Verify that each fold 
is a perpendicular bisector of the other. Then measure the distances to fill 
in the blanks below. 
l(XP) = ........ cm l(XA) = ........ cm l(XB) = ......... cm
l(YP) = ........ cm l(YA) = ........ cm l(YB) = ......... cm
 You will see that all points on the vertical fold are equidistant from the endpoints of 
the horizontal fold.
Try this.
p
q
A
B
M
A
B A
B
P
Q
X
Y
P
A
B
 Fold 1  Fold 2
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FAQs on Textbook: Geometrical Constructions - Mathematics Class 6 (Maharashtra Board)

1. What are geometrical constructions and why are they important in mathematics?
Ans. Geometrical constructions are methods of creating shapes, angles, and figures using only a compass and straightedge. They are important in mathematics because they help develop spatial reasoning, enhance problem-solving skills, and provide a foundation for understanding more complex geometric concepts.
2. What tools are typically used for geometrical constructions?
Ans. The primary tools used for geometrical constructions are a compass and a straightedge (ruler without markings). The compass is used to draw circles or arcs, while the straightedge is used to draw straight lines. These tools allow for precise constructions based solely on geometric principles.
3. Can you explain the basic steps involved in a geometrical construction?
Ans. The basic steps in a geometrical construction typically include: 1. Identifying the construction requirements (e.g., creating a specific angle or bisecting a line). 2. Using the compass to mark points and draw circles or arcs as needed. 3. Connecting points with a straightedge to form the desired geometric figure. 4. Ensuring accuracy by following the steps methodically and checking the results.
4. What are some common geometrical constructions that students learn in Class 6?
Ans. In Class 6, students often learn to construct various figures such as: 1. Bisecting a line segment. 2. Constructing perpendicular bisectors. 3. Creating angles of specific measures (e.g., 60 degrees, 90 degrees). 4. Drawing triangles given certain conditions (e.g., side lengths).
5. How can geometrical constructions be applied in real life?
Ans. Geometrical constructions can be applied in various real-life scenarios, such as architecture and engineering, where precise measurements and shapes are essential. They are also used in art and design to create aesthetically pleasing compositions and in navigation to plot courses based on angles and distances.
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Oct 14, 2025 Last updated
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