Page 1
PRACTICAL GEOMETRY 193 193 193 193 193
10.1 INTRODUCTION
Y ou are familiar with a number of shapes. Y ou learnt how to draw some of them in the earlier
classes. For example, you can draw a line segment of given length, a line perpendicular to a
given line segment, an angle, an angle bisector, a circle etc.
Now, you will learn how to draw parallel lines and some types of triangles.
10.2 CONSTRUCTION OF A LINE PARALLEL TO A GIVEN LINE,
THROUGH A POINT NOT ON THE LINE
Let us begin with an activity (Fig 10.1)
(i) Take a sheet of paper. Make a fold. This
fold represents a line l.
(ii) Unfold the paper. Mark a point A on the
paper outside l.
(iii) Fold the paper perpendicular to the line such
that this perpendicular passes through A.
Name the perpendicular AN.
(iv) Make a fold perpendicular to this
perpendicular through the point A. Name
the new perpendicular line as m. Now, l ||
m. Do you see ‘why’?
Which property or properties of parallel lines
can help you here to say that lines l and m
are parallel.
Chapter 10
Practical
Geometry
(i)
(ii)
(iii)
(iv)
(v)
Fig 10.1
2022-23
Page 2
PRACTICAL GEOMETRY 193 193 193 193 193
10.1 INTRODUCTION
Y ou are familiar with a number of shapes. Y ou learnt how to draw some of them in the earlier
classes. For example, you can draw a line segment of given length, a line perpendicular to a
given line segment, an angle, an angle bisector, a circle etc.
Now, you will learn how to draw parallel lines and some types of triangles.
10.2 CONSTRUCTION OF A LINE PARALLEL TO A GIVEN LINE,
THROUGH A POINT NOT ON THE LINE
Let us begin with an activity (Fig 10.1)
(i) Take a sheet of paper. Make a fold. This
fold represents a line l.
(ii) Unfold the paper. Mark a point A on the
paper outside l.
(iii) Fold the paper perpendicular to the line such
that this perpendicular passes through A.
Name the perpendicular AN.
(iv) Make a fold perpendicular to this
perpendicular through the point A. Name
the new perpendicular line as m. Now, l ||
m. Do you see ‘why’?
Which property or properties of parallel lines
can help you here to say that lines l and m
are parallel.
Chapter 10
Practical
Geometry
(i)
(ii)
(iii)
(iv)
(v)
Fig 10.1
2022-23
MATHEMATICS 194 194 194 194 194
Y ou can use any one of the properties regarding the transversal and parallel lines to
make this construction using ruler and compasses only.
Step 1 Take a line ‘l ’ and a point ‘A ’ outside ‘l ’ [Fig10.2 (i)].
Step 2 Take any point B on l and join B to A [Fig 10.2(ii)].
Step 3 With B as centre and a convenient radius, draw an arc cutting l at C and BA at D
[Fig 10.2(iii)].
Step 4 Now with A as centre and the same radius as in Step 3, draw an arc EF cutting AB
at G [Fig 10.2 (iv)].
2022-23
Page 3
PRACTICAL GEOMETRY 193 193 193 193 193
10.1 INTRODUCTION
Y ou are familiar with a number of shapes. Y ou learnt how to draw some of them in the earlier
classes. For example, you can draw a line segment of given length, a line perpendicular to a
given line segment, an angle, an angle bisector, a circle etc.
Now, you will learn how to draw parallel lines and some types of triangles.
10.2 CONSTRUCTION OF A LINE PARALLEL TO A GIVEN LINE,
THROUGH A POINT NOT ON THE LINE
Let us begin with an activity (Fig 10.1)
(i) Take a sheet of paper. Make a fold. This
fold represents a line l.
(ii) Unfold the paper. Mark a point A on the
paper outside l.
(iii) Fold the paper perpendicular to the line such
that this perpendicular passes through A.
Name the perpendicular AN.
(iv) Make a fold perpendicular to this
perpendicular through the point A. Name
the new perpendicular line as m. Now, l ||
m. Do you see ‘why’?
Which property or properties of parallel lines
can help you here to say that lines l and m
are parallel.
Chapter 10
Practical
Geometry
(i)
(ii)
(iii)
(iv)
(v)
Fig 10.1
2022-23
MATHEMATICS 194 194 194 194 194
Y ou can use any one of the properties regarding the transversal and parallel lines to
make this construction using ruler and compasses only.
Step 1 Take a line ‘l ’ and a point ‘A ’ outside ‘l ’ [Fig10.2 (i)].
Step 2 Take any point B on l and join B to A [Fig 10.2(ii)].
Step 3 With B as centre and a convenient radius, draw an arc cutting l at C and BA at D
[Fig 10.2(iii)].
Step 4 Now with A as centre and the same radius as in Step 3, draw an arc EF cutting AB
at G [Fig 10.2 (iv)].
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PRACTICAL GEOMETRY 195 195 195 195 195
Step 5 Place the pointed tip of the compasses at C and adjust the opening so that the
pencil tip is at D [Fig 10.2 (v)].
Step 6 With the same opening as in Step 5 and with G as centre, draw an arc cutting the
arc EF at H [Fig 10.2 (vi)].
Step 7 Now, join AH to draw a line ‘m’ [Fig 10.2 (vii)].
Note that ?ABC and ?BAH are alternate interior angles.
Therefore m || l
THINK, DISCUSS AND WRITE
1. In the above construction, can you draw any other line through A that would be also
parallel to the line l ?
2. Can you slightly modify the above construction to use the idea of equal corresponding
angles instead of equal alternate angles?
Fig 10.2 (i)–(vii)
2022-23
Page 4
PRACTICAL GEOMETRY 193 193 193 193 193
10.1 INTRODUCTION
Y ou are familiar with a number of shapes. Y ou learnt how to draw some of them in the earlier
classes. For example, you can draw a line segment of given length, a line perpendicular to a
given line segment, an angle, an angle bisector, a circle etc.
Now, you will learn how to draw parallel lines and some types of triangles.
10.2 CONSTRUCTION OF A LINE PARALLEL TO A GIVEN LINE,
THROUGH A POINT NOT ON THE LINE
Let us begin with an activity (Fig 10.1)
(i) Take a sheet of paper. Make a fold. This
fold represents a line l.
(ii) Unfold the paper. Mark a point A on the
paper outside l.
(iii) Fold the paper perpendicular to the line such
that this perpendicular passes through A.
Name the perpendicular AN.
(iv) Make a fold perpendicular to this
perpendicular through the point A. Name
the new perpendicular line as m. Now, l ||
m. Do you see ‘why’?
Which property or properties of parallel lines
can help you here to say that lines l and m
are parallel.
Chapter 10
Practical
Geometry
(i)
(ii)
(iii)
(iv)
(v)
Fig 10.1
2022-23
MATHEMATICS 194 194 194 194 194
Y ou can use any one of the properties regarding the transversal and parallel lines to
make this construction using ruler and compasses only.
Step 1 Take a line ‘l ’ and a point ‘A ’ outside ‘l ’ [Fig10.2 (i)].
Step 2 Take any point B on l and join B to A [Fig 10.2(ii)].
Step 3 With B as centre and a convenient radius, draw an arc cutting l at C and BA at D
[Fig 10.2(iii)].
Step 4 Now with A as centre and the same radius as in Step 3, draw an arc EF cutting AB
at G [Fig 10.2 (iv)].
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PRACTICAL GEOMETRY 195 195 195 195 195
Step 5 Place the pointed tip of the compasses at C and adjust the opening so that the
pencil tip is at D [Fig 10.2 (v)].
Step 6 With the same opening as in Step 5 and with G as centre, draw an arc cutting the
arc EF at H [Fig 10.2 (vi)].
Step 7 Now, join AH to draw a line ‘m’ [Fig 10.2 (vii)].
Note that ?ABC and ?BAH are alternate interior angles.
Therefore m || l
THINK, DISCUSS AND WRITE
1. In the above construction, can you draw any other line through A that would be also
parallel to the line l ?
2. Can you slightly modify the above construction to use the idea of equal corresponding
angles instead of equal alternate angles?
Fig 10.2 (i)–(vii)
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MATHEMATICS 196 196 196 196 196
EXERCISE 10.1
1. Draw a line, say AB, take a point C outside it. Through C, draw a line parallel to AB
using ruler and compasses only.
2. Draw a line l. Draw a perpendicular to l at any point on l. On this perpendicular
choose a point X, 4 cm away from l. Through X, draw a line m parallel to l.
3. Let l be a line and P be a point not on l. Through P , draw a line m parallel to l. Now
join P to any point Q on l. Choose any other point R on m. Through R, draw a line
parallel to PQ. Let this meet l at S. What shape do the two sets of parallel lines enclose?
10.3 CONSTRUCTION OF TRIANGLES
It is better for you to go through this section after recalling ideas
on triangles, in particular, the chapters on properties of triangles
and congruence of triangles.
You know how triangles are classified based on sides or
angles and the following important properties concerning triangles:
(i) The exterior angle of a triangle is equal in measure to the
sum of interior opposite angles.
(ii) The total measure of the three angles of a triangle is 180°.
(iii) Sum of the lengths of any two sides of a triangle is greater
than the length of the third side.
(iv) In any right-angled triangle, the square of the length of
hypotenuse is equal to the sum of the squares of the lengths of the other two
sides.
In the chapter on ‘Congruence of Triangles’, we saw that a triangle can be drawn if any
one of the following sets of measurements are given:
(i) Three sides.
(ii) T wo sides and the angle between them.
(iii) T wo angles and the side between them.
(iv) The hypotenuse and a leg in the case of a right-angled triangle.
We will now attempt to use these ideas to construct triangles.
10.4 CONSTRUCTING A TRIANGLE WHEN THE LENGTHS OF
ITS THREE SIDES ARE KNOWN (SSS CRITERION)
In this section, we would construct triangles when all its sides are known. We draw first a
rough sketch to give an idea of where the sides are and then begin by drawing any one of
?3 = ?1 + ?2
a+ b > c
?1 + ?2 + ?3 = 180°
b
2
+ a
2
= c
2
2022-23
Page 5
PRACTICAL GEOMETRY 193 193 193 193 193
10.1 INTRODUCTION
Y ou are familiar with a number of shapes. Y ou learnt how to draw some of them in the earlier
classes. For example, you can draw a line segment of given length, a line perpendicular to a
given line segment, an angle, an angle bisector, a circle etc.
Now, you will learn how to draw parallel lines and some types of triangles.
10.2 CONSTRUCTION OF A LINE PARALLEL TO A GIVEN LINE,
THROUGH A POINT NOT ON THE LINE
Let us begin with an activity (Fig 10.1)
(i) Take a sheet of paper. Make a fold. This
fold represents a line l.
(ii) Unfold the paper. Mark a point A on the
paper outside l.
(iii) Fold the paper perpendicular to the line such
that this perpendicular passes through A.
Name the perpendicular AN.
(iv) Make a fold perpendicular to this
perpendicular through the point A. Name
the new perpendicular line as m. Now, l ||
m. Do you see ‘why’?
Which property or properties of parallel lines
can help you here to say that lines l and m
are parallel.
Chapter 10
Practical
Geometry
(i)
(ii)
(iii)
(iv)
(v)
Fig 10.1
2022-23
MATHEMATICS 194 194 194 194 194
Y ou can use any one of the properties regarding the transversal and parallel lines to
make this construction using ruler and compasses only.
Step 1 Take a line ‘l ’ and a point ‘A ’ outside ‘l ’ [Fig10.2 (i)].
Step 2 Take any point B on l and join B to A [Fig 10.2(ii)].
Step 3 With B as centre and a convenient radius, draw an arc cutting l at C and BA at D
[Fig 10.2(iii)].
Step 4 Now with A as centre and the same radius as in Step 3, draw an arc EF cutting AB
at G [Fig 10.2 (iv)].
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PRACTICAL GEOMETRY 195 195 195 195 195
Step 5 Place the pointed tip of the compasses at C and adjust the opening so that the
pencil tip is at D [Fig 10.2 (v)].
Step 6 With the same opening as in Step 5 and with G as centre, draw an arc cutting the
arc EF at H [Fig 10.2 (vi)].
Step 7 Now, join AH to draw a line ‘m’ [Fig 10.2 (vii)].
Note that ?ABC and ?BAH are alternate interior angles.
Therefore m || l
THINK, DISCUSS AND WRITE
1. In the above construction, can you draw any other line through A that would be also
parallel to the line l ?
2. Can you slightly modify the above construction to use the idea of equal corresponding
angles instead of equal alternate angles?
Fig 10.2 (i)–(vii)
2022-23
MATHEMATICS 196 196 196 196 196
EXERCISE 10.1
1. Draw a line, say AB, take a point C outside it. Through C, draw a line parallel to AB
using ruler and compasses only.
2. Draw a line l. Draw a perpendicular to l at any point on l. On this perpendicular
choose a point X, 4 cm away from l. Through X, draw a line m parallel to l.
3. Let l be a line and P be a point not on l. Through P , draw a line m parallel to l. Now
join P to any point Q on l. Choose any other point R on m. Through R, draw a line
parallel to PQ. Let this meet l at S. What shape do the two sets of parallel lines enclose?
10.3 CONSTRUCTION OF TRIANGLES
It is better for you to go through this section after recalling ideas
on triangles, in particular, the chapters on properties of triangles
and congruence of triangles.
You know how triangles are classified based on sides or
angles and the following important properties concerning triangles:
(i) The exterior angle of a triangle is equal in measure to the
sum of interior opposite angles.
(ii) The total measure of the three angles of a triangle is 180°.
(iii) Sum of the lengths of any two sides of a triangle is greater
than the length of the third side.
(iv) In any right-angled triangle, the square of the length of
hypotenuse is equal to the sum of the squares of the lengths of the other two
sides.
In the chapter on ‘Congruence of Triangles’, we saw that a triangle can be drawn if any
one of the following sets of measurements are given:
(i) Three sides.
(ii) T wo sides and the angle between them.
(iii) T wo angles and the side between them.
(iv) The hypotenuse and a leg in the case of a right-angled triangle.
We will now attempt to use these ideas to construct triangles.
10.4 CONSTRUCTING A TRIANGLE WHEN THE LENGTHS OF
ITS THREE SIDES ARE KNOWN (SSS CRITERION)
In this section, we would construct triangles when all its sides are known. We draw first a
rough sketch to give an idea of where the sides are and then begin by drawing any one of
?3 = ?1 + ?2
a+ b > c
?1 + ?2 + ?3 = 180°
b
2
+ a
2
= c
2
2022-23
PRACTICAL GEOMETRY 197 197 197 197 197
the three lines. See the following example:
EXAMPLE 1 Construct a triangle ABC, given that AB = 5 cm, BC = 6 cm and AC = 7 cm.
SOLUTION
Step 1 First, we draw a rough sketch with given measure, (This will help us in
deciding how to proceed) [Fig 10.3(i)].
Step 2 Draw a line segment BC of length 6 cm [Fig 10.3(ii)].
Step 3 From B, point A is at a distance of 5 cm. So, with B as centre, draw an arc of
radius 5 cm. (Now A will be somewhere on this arc. Our job is to find where
exactly A is) [Fig 10.3(iii)].
Step 4 From C, point A is at a distance of 7 cm. So, with C as centre, draw an arc of
radius 7 cm. (A will be somewhere on this arc, we have to fix it) [Fig 10.3(iv)].
(ii)
(i)
(iii)
(iv)
(Rough Sketch)
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