RS Aggarwal Solutions: Two-Dimensional Reflection Symmetry (Linear Symmetry) | Mathematics (Maths) Class 6 PDF Download

 Page 1


     
Points to Remember :
1. Line symmetry. A figure is said to be symmetrical about a line l, if it is identical on either side
of l. Line l is known as the line of symmetry or axis of symmetry.
2. Some examples of Linear Symmetry.
Example 1. A line segment is symmetrical about its perpendicular bisector .
Method. Let AB be a given line segment and let POQ be the perpendicular bisector of AB.
Hence, the line segment AB is symmetrical about its perpendicular bisector POQ.
Example 2. A given angle having equal arms is symmetrical about the bisector of the angle.
Method. Let AOB be a given angle with equal arms OA and OB, and let OC be the bisector of AOB.
Then clearly AOC and BOC are identical.
Hence, AOB is symmetrical about the bisector OC.
Example 3. An isosceles triangle is symmetrical about the bisector of the angle included between the
equal sides.
Method. Let ABC be an isosceles triangle in which AB = AC and let AD be the bisector of BAC.
Page 2


     
Points to Remember :
1. Line symmetry. A figure is said to be symmetrical about a line l, if it is identical on either side
of l. Line l is known as the line of symmetry or axis of symmetry.
2. Some examples of Linear Symmetry.
Example 1. A line segment is symmetrical about its perpendicular bisector .
Method. Let AB be a given line segment and let POQ be the perpendicular bisector of AB.
Hence, the line segment AB is symmetrical about its perpendicular bisector POQ.
Example 2. A given angle having equal arms is symmetrical about the bisector of the angle.
Method. Let AOB be a given angle with equal arms OA and OB, and let OC be the bisector of AOB.
Then clearly AOC and BOC are identical.
Hence, AOB is symmetrical about the bisector OC.
Example 3. An isosceles triangle is symmetrical about the bisector of the angle included between the
equal sides.
Method. Let ABC be an isosceles triangle in which AB = AC and let AD be the bisector of BAC.
If ABC be folded along AD
then ADC coincides exactly
with ADB.
Thus, ADC is identical
with ADB.
Hence, AD is the line
of symmetry of ABC.
Example 4. A kite has one line of symmetry, namely
the diagonal shown dotted in the adjoining figure
Method. Here ABCD is a kite
in which AB = AD and BC = DC.
If we fold the kite along the
line AC, we find that the two
parts coincide with each other.
Hence, the kite ABCD is
symmetrical about the diagonal AC.
Example 5. A semicircle ACB has one
line of symmetry, namely the perpendi- cular
bisector of the diameter AB.
Method. Here, ACB is a semicircle and
PQ is the perpendicular bisector of diameter AB.
If we fold the semicircle along the line PQ, we find
that the two parts of it coincide with each other.
Hence, the semicircle ACB is
symmetrical about the perpendicular
bisector of diameter AB.
Example 6. An isosceles trapezium has
one line of symmetry, namely the line joining the
midpoints of the bases of the trapezium.
Method. Let ABCD be an isosceles
trapezium in which AB || DC and AD = BC.
Let E and F be the midpoints of AB and
DC respectively.
If we fold the trapezium along the line
EF, we find that the two parts of it coincide with
each other.
Hence, the trapezium ABCD is
symmetrical about the line EF.
Example 7. A rectangle has two lines of
symmetry, each one of which being the line joining
the midpoints of opposite sides.
Method. Let ABCD be a given rectangle,
and let P and Q be the midpoints of AB and DC
respectively.
Page 3


     
Points to Remember :
1. Line symmetry. A figure is said to be symmetrical about a line l, if it is identical on either side
of l. Line l is known as the line of symmetry or axis of symmetry.
2. Some examples of Linear Symmetry.
Example 1. A line segment is symmetrical about its perpendicular bisector .
Method. Let AB be a given line segment and let POQ be the perpendicular bisector of AB.
Hence, the line segment AB is symmetrical about its perpendicular bisector POQ.
Example 2. A given angle having equal arms is symmetrical about the bisector of the angle.
Method. Let AOB be a given angle with equal arms OA and OB, and let OC be the bisector of AOB.
Then clearly AOC and BOC are identical.
Hence, AOB is symmetrical about the bisector OC.
Example 3. An isosceles triangle is symmetrical about the bisector of the angle included between the
equal sides.
Method. Let ABC be an isosceles triangle in which AB = AC and let AD be the bisector of BAC.
If ABC be folded along AD
then ADC coincides exactly
with ADB.
Thus, ADC is identical
with ADB.
Hence, AD is the line
of symmetry of ABC.
Example 4. A kite has one line of symmetry, namely
the diagonal shown dotted in the adjoining figure
Method. Here ABCD is a kite
in which AB = AD and BC = DC.
If we fold the kite along the
line AC, we find that the two
parts coincide with each other.
Hence, the kite ABCD is
symmetrical about the diagonal AC.
Example 5. A semicircle ACB has one
line of symmetry, namely the perpendi- cular
bisector of the diameter AB.
Method. Here, ACB is a semicircle and
PQ is the perpendicular bisector of diameter AB.
If we fold the semicircle along the line PQ, we find
that the two parts of it coincide with each other.
Hence, the semicircle ACB is
symmetrical about the perpendicular
bisector of diameter AB.
Example 6. An isosceles trapezium has
one line of symmetry, namely the line joining the
midpoints of the bases of the trapezium.
Method. Let ABCD be an isosceles
trapezium in which AB || DC and AD = BC.
Let E and F be the midpoints of AB and
DC respectively.
If we fold the trapezium along the line
EF, we find that the two parts of it coincide with
each other.
Hence, the trapezium ABCD is
symmetrical about the line EF.
Example 7. A rectangle has two lines of
symmetry, each one of which being the line joining
the midpoints of opposite sides.
Method. Let ABCD be a given rectangle,
and let P and Q be the midpoints of AB and DC
respectively.
Now, if we fold the rectangle along PQ, we find
that the two parts of it coincide with each other.
Hence, rectangle ABCD is symmetrical
about the line PQ.
Similarly, if R and S be the midpoints of
AD and BC respectively then rectangle ABCD is
symmetrical about the line RS.
Example 8. A rhombus is symmetrical
about each one of its diagonals.
Method. Let ABCD be a rhombus. Now,
if we fold it along the diagonal AC, we find that
the two parts coincide with each other.
Hence, the rhombus ABCD is symmetrical
about its diagonal AC.
Similarly, the rhombus ABCD is
symmetrical about its diagonal BD.
Example 9. A square has four lines of
symmetry, namely the diagonals and the lines
joining the midpoints of its opposite sides.
Method. Let ABCD be the given square
and E, F, G, H be the mid-points of AB, DC, AD
and BC respectively.
Then, it is easy to see that it is symmetrical about
each of the lines AC, BD, EF and GH.
Example 10. An equilateral triangle is symmetrical
about each one of the bisectors of its interior angles.
Method. Let ABC be an equilateral triangle and
let AD, BE and CF be the bisectors of A, B
and C respectively.
Then, it is easy to see that ABC is symmetrical
about each of the lines AD, BE and CF.
Example 11. A circle is symmetrical about
each of its diameters. Thus, each diameter of a
circle is an axis of symmetry.
Method. Here, a number of diameters of
a circle have been drawn. It is easy to see that the
circle is symmetrical about each of the diameters
drawn. Hence, a circle has an infinite number of
lines of symmetry.
Remarks. (i) A scalene triangle has no
line of symmetry.
Page 4


     
Points to Remember :
1. Line symmetry. A figure is said to be symmetrical about a line l, if it is identical on either side
of l. Line l is known as the line of symmetry or axis of symmetry.
2. Some examples of Linear Symmetry.
Example 1. A line segment is symmetrical about its perpendicular bisector .
Method. Let AB be a given line segment and let POQ be the perpendicular bisector of AB.
Hence, the line segment AB is symmetrical about its perpendicular bisector POQ.
Example 2. A given angle having equal arms is symmetrical about the bisector of the angle.
Method. Let AOB be a given angle with equal arms OA and OB, and let OC be the bisector of AOB.
Then clearly AOC and BOC are identical.
Hence, AOB is symmetrical about the bisector OC.
Example 3. An isosceles triangle is symmetrical about the bisector of the angle included between the
equal sides.
Method. Let ABC be an isosceles triangle in which AB = AC and let AD be the bisector of BAC.
If ABC be folded along AD
then ADC coincides exactly
with ADB.
Thus, ADC is identical
with ADB.
Hence, AD is the line
of symmetry of ABC.
Example 4. A kite has one line of symmetry, namely
the diagonal shown dotted in the adjoining figure
Method. Here ABCD is a kite
in which AB = AD and BC = DC.
If we fold the kite along the
line AC, we find that the two
parts coincide with each other.
Hence, the kite ABCD is
symmetrical about the diagonal AC.
Example 5. A semicircle ACB has one
line of symmetry, namely the perpendi- cular
bisector of the diameter AB.
Method. Here, ACB is a semicircle and
PQ is the perpendicular bisector of diameter AB.
If we fold the semicircle along the line PQ, we find
that the two parts of it coincide with each other.
Hence, the semicircle ACB is
symmetrical about the perpendicular
bisector of diameter AB.
Example 6. An isosceles trapezium has
one line of symmetry, namely the line joining the
midpoints of the bases of the trapezium.
Method. Let ABCD be an isosceles
trapezium in which AB || DC and AD = BC.
Let E and F be the midpoints of AB and
DC respectively.
If we fold the trapezium along the line
EF, we find that the two parts of it coincide with
each other.
Hence, the trapezium ABCD is
symmetrical about the line EF.
Example 7. A rectangle has two lines of
symmetry, each one of which being the line joining
the midpoints of opposite sides.
Method. Let ABCD be a given rectangle,
and let P and Q be the midpoints of AB and DC
respectively.
Now, if we fold the rectangle along PQ, we find
that the two parts of it coincide with each other.
Hence, rectangle ABCD is symmetrical
about the line PQ.
Similarly, if R and S be the midpoints of
AD and BC respectively then rectangle ABCD is
symmetrical about the line RS.
Example 8. A rhombus is symmetrical
about each one of its diagonals.
Method. Let ABCD be a rhombus. Now,
if we fold it along the diagonal AC, we find that
the two parts coincide with each other.
Hence, the rhombus ABCD is symmetrical
about its diagonal AC.
Similarly, the rhombus ABCD is
symmetrical about its diagonal BD.
Example 9. A square has four lines of
symmetry, namely the diagonals and the lines
joining the midpoints of its opposite sides.
Method. Let ABCD be the given square
and E, F, G, H be the mid-points of AB, DC, AD
and BC respectively.
Then, it is easy to see that it is symmetrical about
each of the lines AC, BD, EF and GH.
Example 10. An equilateral triangle is symmetrical
about each one of the bisectors of its interior angles.
Method. Let ABC be an equilateral triangle and
let AD, BE and CF be the bisectors of A, B
and C respectively.
Then, it is easy to see that ABC is symmetrical
about each of the lines AD, BE and CF.
Example 11. A circle is symmetrical about
each of its diameters. Thus, each diameter of a
circle is an axis of symmetry.
Method. Here, a number of diameters of
a circle have been drawn. It is easy to see that the
circle is symmetrical about each of the diameters
drawn. Hence, a circle has an infinite number of
lines of symmetry.
Remarks. (i) A scalene triangle has no
line of symmetry.
(ii) A parallelogram has no line of symmetry.
Example 12. Each of the following capital letters
of the English alphabet is symmetrical about the
dotted line or lines as shown.
EXERCISE 20
Mark ( ) against the correct answer
in each of Q. 1 to Q. 8.
Q. 1. A square has
(a) one line of symmetry (b) two lines of symmetry
(c) three lines of symmetry
(d) four lines of symmetry
Sol. (d) 
.
.
.
 A square has four lines of
symmetry, two diagonals and two lines
joining the mid-points of opposite sides.
Q. 2. A rectangle is symmetrical about
(a) each one of its sides
 (b) each one of its diagonals
(c) a line joining the midpoints of its opposite sides
(d) none of these
Sol. (c) 
.
.
.
 A rectangle has two lines of
symmetry, each one of which being the line joining
of mid-points of opposite sides.
Q. 3. A rhombus is symmetrical about
(a) the line joining the midpoints of its opposite sides
(b) each of its diagonals
(c) perpendicular bisector of each of its sides
(d) none of these
Sol. (b) 
.
.
.
 A rhombus has two lines of symmetry
namely two diagonals.
Q. 4. A circle has
(a) no line of symmetry
(b) one line of symmetry
(c) two lines of symmetry
(d) an unlimited number of lines of
symmetry.
Sol. (d) 
.
.
.
 Each diameter of a circle is its line
of symmetry which are unlimited
numbers.
Q. 5. A scalene triangle has
(a) no line of symmetry (b) one line of symmetry
(c) two lines of symmetry (d) three lines of symmetry
Sol. (a) 
.
.
.
 A scalene triangle has no line of
symmetry.
Q. 6. ABCD is a kite in which AB = AD and BC
= DC.
 The kite is symmetrical about
 (a) the diagonal AC
 (b) the diagonal BD
(c) none of these
Sol. (a) 
.
.
.
 It is a figure of kite ; so AC is its line of
symmetry.
Q. 7. The letter O of the English alphabet has
(a) no line of symmetry  (b) one line of symmetry
(c) two lines of symmetry  (d) none of these
Sol. (c) 
.
.
.
 Letter O has two lines of symmetry,
one vertical and second horizontal.
Q. 8. The letter Z of the English alphabet has
(a) no line of symmetry (b) one line of symmetry
Page 5


     
Points to Remember :
1. Line symmetry. A figure is said to be symmetrical about a line l, if it is identical on either side
of l. Line l is known as the line of symmetry or axis of symmetry.
2. Some examples of Linear Symmetry.
Example 1. A line segment is symmetrical about its perpendicular bisector .
Method. Let AB be a given line segment and let POQ be the perpendicular bisector of AB.
Hence, the line segment AB is symmetrical about its perpendicular bisector POQ.
Example 2. A given angle having equal arms is symmetrical about the bisector of the angle.
Method. Let AOB be a given angle with equal arms OA and OB, and let OC be the bisector of AOB.
Then clearly AOC and BOC are identical.
Hence, AOB is symmetrical about the bisector OC.
Example 3. An isosceles triangle is symmetrical about the bisector of the angle included between the
equal sides.
Method. Let ABC be an isosceles triangle in which AB = AC and let AD be the bisector of BAC.
If ABC be folded along AD
then ADC coincides exactly
with ADB.
Thus, ADC is identical
with ADB.
Hence, AD is the line
of symmetry of ABC.
Example 4. A kite has one line of symmetry, namely
the diagonal shown dotted in the adjoining figure
Method. Here ABCD is a kite
in which AB = AD and BC = DC.
If we fold the kite along the
line AC, we find that the two
parts coincide with each other.
Hence, the kite ABCD is
symmetrical about the diagonal AC.
Example 5. A semicircle ACB has one
line of symmetry, namely the perpendi- cular
bisector of the diameter AB.
Method. Here, ACB is a semicircle and
PQ is the perpendicular bisector of diameter AB.
If we fold the semicircle along the line PQ, we find
that the two parts of it coincide with each other.
Hence, the semicircle ACB is
symmetrical about the perpendicular
bisector of diameter AB.
Example 6. An isosceles trapezium has
one line of symmetry, namely the line joining the
midpoints of the bases of the trapezium.
Method. Let ABCD be an isosceles
trapezium in which AB || DC and AD = BC.
Let E and F be the midpoints of AB and
DC respectively.
If we fold the trapezium along the line
EF, we find that the two parts of it coincide with
each other.
Hence, the trapezium ABCD is
symmetrical about the line EF.
Example 7. A rectangle has two lines of
symmetry, each one of which being the line joining
the midpoints of opposite sides.
Method. Let ABCD be a given rectangle,
and let P and Q be the midpoints of AB and DC
respectively.
Now, if we fold the rectangle along PQ, we find
that the two parts of it coincide with each other.
Hence, rectangle ABCD is symmetrical
about the line PQ.
Similarly, if R and S be the midpoints of
AD and BC respectively then rectangle ABCD is
symmetrical about the line RS.
Example 8. A rhombus is symmetrical
about each one of its diagonals.
Method. Let ABCD be a rhombus. Now,
if we fold it along the diagonal AC, we find that
the two parts coincide with each other.
Hence, the rhombus ABCD is symmetrical
about its diagonal AC.
Similarly, the rhombus ABCD is
symmetrical about its diagonal BD.
Example 9. A square has four lines of
symmetry, namely the diagonals and the lines
joining the midpoints of its opposite sides.
Method. Let ABCD be the given square
and E, F, G, H be the mid-points of AB, DC, AD
and BC respectively.
Then, it is easy to see that it is symmetrical about
each of the lines AC, BD, EF and GH.
Example 10. An equilateral triangle is symmetrical
about each one of the bisectors of its interior angles.
Method. Let ABC be an equilateral triangle and
let AD, BE and CF be the bisectors of A, B
and C respectively.
Then, it is easy to see that ABC is symmetrical
about each of the lines AD, BE and CF.
Example 11. A circle is symmetrical about
each of its diameters. Thus, each diameter of a
circle is an axis of symmetry.
Method. Here, a number of diameters of
a circle have been drawn. It is easy to see that the
circle is symmetrical about each of the diameters
drawn. Hence, a circle has an infinite number of
lines of symmetry.
Remarks. (i) A scalene triangle has no
line of symmetry.
(ii) A parallelogram has no line of symmetry.
Example 12. Each of the following capital letters
of the English alphabet is symmetrical about the
dotted line or lines as shown.
EXERCISE 20
Mark ( ) against the correct answer
in each of Q. 1 to Q. 8.
Q. 1. A square has
(a) one line of symmetry (b) two lines of symmetry
(c) three lines of symmetry
(d) four lines of symmetry
Sol. (d) 
.
.
.
 A square has four lines of
symmetry, two diagonals and two lines
joining the mid-points of opposite sides.
Q. 2. A rectangle is symmetrical about
(a) each one of its sides
 (b) each one of its diagonals
(c) a line joining the midpoints of its opposite sides
(d) none of these
Sol. (c) 
.
.
.
 A rectangle has two lines of
symmetry, each one of which being the line joining
of mid-points of opposite sides.
Q. 3. A rhombus is symmetrical about
(a) the line joining the midpoints of its opposite sides
(b) each of its diagonals
(c) perpendicular bisector of each of its sides
(d) none of these
Sol. (b) 
.
.
.
 A rhombus has two lines of symmetry
namely two diagonals.
Q. 4. A circle has
(a) no line of symmetry
(b) one line of symmetry
(c) two lines of symmetry
(d) an unlimited number of lines of
symmetry.
Sol. (d) 
.
.
.
 Each diameter of a circle is its line
of symmetry which are unlimited
numbers.
Q. 5. A scalene triangle has
(a) no line of symmetry (b) one line of symmetry
(c) two lines of symmetry (d) three lines of symmetry
Sol. (a) 
.
.
.
 A scalene triangle has no line of
symmetry.
Q. 6. ABCD is a kite in which AB = AD and BC
= DC.
 The kite is symmetrical about
 (a) the diagonal AC
 (b) the diagonal BD
(c) none of these
Sol. (a) 
.
.
.
 It is a figure of kite ; so AC is its line of
symmetry.
Q. 7. The letter O of the English alphabet has
(a) no line of symmetry  (b) one line of symmetry
(c) two lines of symmetry  (d) none of these
Sol. (c) 
.
.
.
 Letter O has two lines of symmetry,
one vertical and second horizontal.
Q. 8. The letter Z of the English alphabet has
(a) no line of symmetry (b) one line of symmetry
(c) two lines of symmetry (d) none of these
Sol. (a) 
.
.
.
 Letter Z has no line of symmetry.
Q. 9. Draw the line (or lines) of symmetry of each of the following figures.
Sol.
Q. 10. Which of the following statements are true and which are False ?
(i) A parallelogram has no line of symmetry.
(ii) An angle with equal arms has its bisector as the line of symmetry.
(iii) An equilateral triangle has three lines of symmetry.
(iv) A rhombus has four lines of symmetry.
(v) A square has four lines of symmetry.
(vi) A rectangle has two lines of symmetry.
(vii) Each one of the letters H, I, O, X of the English alphabet has two lines of symmetry.
Sol. (i) True (T) 
.
.
.
 Parallelogram has no line of symmetry.
(ii) True (T) 
.
.
.
 Bisector of an angle of equal sides is the line of symmetry.
(iii) True (T) 
.
.
.
 Perpendiculars from each vertices of an equilateral triangle to its opposite side is its
line of symmetry.
(iv) False (F) 
.
.
.
 Rhombus has two lines of symmetry which are its diagonals.
(v) True (T) 
.
.
.
 Square has four lines of symmetry, two diagonals and two perpendicular bisectors
of opposite sides.
(vi) True (T) 
.
.
.
 A rectangle has two lines of symmetry which are the perpendicular bisectors of its
opposite sides.
(vii) True (T) 
.
.
.
 H, I, O and X has two lines of symmetry.