Which of the following figures has only one line of symmetry?
(i) A rectangle
(ii) A parallelogram
(iii) An isosceles trapezium
(iv) A circle
Answer
(i) A rectangle
Rectangle has two lines of symmetry.
These are the lines that join the midpoints of the opposite sides.
(ii) A parallelogram
A parallelogram has no lines of symmetry.
Even though it looks balanced, if we fold it along any line, the vertices will not align.
(iii) An isosceles trapezium
An isosceles trapezium has exactly one line of symmetry.
This line passes through the midpoints of the two parallel sides. This vertical fold ensures the two non-parallel equal sides sit perfectly on top of each other.
(iv) A circle
A circle is perfectly symmetrical.
Any line passing through the centre acts as a line of symmetry. Since we can draw an infinite number of diameters, there are infinitely many lines of symmetry.
∴ An isosceles trapezium has only one line of symmetry.
An angle having equal arms possesses how many lines of symmetry? Show all possible lines of symmetry in such an angle.
Answer
An angle with equal arms possesses one line of symmetry.
The line of symmetry is the angle bisector.
This is the line that passes through the vertex of the angle and divides it into two equal parts. If we fold the angle along this bisector, the two arms will coincide perfectly.
∴ One line of symmetry, i.e., its angle bisector.
How many lines of symmetry does a scalene triangle have?
Answer
A scalene triangle has three sides of different lengths and three angles of different measures. Because no two sides or angles are equal, there is no way to draw a line through the triangle that creates two identical, mirror-image halves.
∴ A scalene triangle has no line of symmetry.
A square and a rectangle have :
only one line of symmetry each
two lines of symmetry each
four lines of symmetry each
an unequal number of lines of symmetry
Answer
A square has 4 lines of symmetry: 2 joining the midpoints of opposite sides and 2 along the diagonals.
A rectangle has 2 lines of symmetry, i.e., the lines joining the midpoints of opposite sides.
Since 4 is not equal to 2, they have an unequal number of lines of symmetry.
∴ Option 4 is the correct option.
A rhombus has :
one line of symmetry
two lines of symmetry
four lines of symmetry
no line of symmetry
Answer
A rhombus has 2 lines of symmetry. These lines are its diagonals.
If we fold a rhombus along either diagonal, the opposite vertices and sides will match perfectly. Unlike a square, the lines joining the midpoints of the sides of a rhombus are not lines of symmetry.
∴ Option 2 is the correct option.
A parallelogram has :
no line of symmetry
one line of symmetry
the same number of lines of symmetry as the rhombus
four lines of symmetry
Answer
A general parallelogram has no lines of symmetry.
While it has rotational symmetry (it looks the same if we rotate it 180°), there is no line along which we can fold it to make the two halves coincide.
∴ Option 1 is the correct option.
The capital letter S in the English alphabet has :
no line of symmetry
one line of symmetry
two lines of symmetry
infinite number of lines of symmetry
Answer
The letter S has no line of symmetry.
If you draw a horizontal line through the middle, the top curve points one way and the bottom curve points the other.
If you draw a vertical line, the two sides do not mirror each other.
∴ Option 1 is the correct option.
Draw all possible lines of symmetry in each of the following figures and state the number of lines of symmetry in each case.
(i)
(ii)
(iii)
(iv)
Answer
(i)
Lines of symmetry = 1
(ii)
Lines of symmetry = 1
(iii)
Lines of symmetry = 2
(iv)
Lines of symmetry = 4
Draw a regular octagon and draw all possible lines of symmetry in it.
Answer
∴ A regular octagon has 8 lines of symmetry.
Draw all possible lines of symmetry in :
(i) a regular pentagon
(ii) a regular hexagon
Answer
(i) a regular pentagon
∴ A regular pentagon has 5 lines of symmetry
(ii) a regular hexagon
∴ A regular hexagon has 6 lines of symmetry
Construct a triangle ABC such that BC = 6.5 cm and ∠B = ∠C = 70°. Draw all possible lines of symmetry.
Answer
This is an isosceles triangle because two angles are equal.
∴ It has 1 line of symmetry.
Construct a triangle ABC having BC = 5 cm, ∠B = 90° and ∠C = 45°. Draw all possible lines of symmetry.
Answer
∠A = 180° - (90° + 45°) = 45°.
Since ∠A = ∠C = 45°, this is an isosceles right-angled triangle.
∴ It has 1 line of symmetry.
Construct a triangle PQR having PQ = 6 cm, ∠P = ∠R = 60° and draw all possible lines of symmetry.
Answer
∠Q = 180° - (60° + 60°) = 60°.
Since all angles are 60°, this is an equilateral triangle.
∴ It has 3 lines of symmetry.
Construct a square having each side equal to 4 cm. Draw all possible lines of symmetry.
Answer
∴ A square has exactly 4 lines of symmetry.
Construct a straight line AB = 7.3 cm. Draw its line of symmetry (using a pair of compasses).
Answer
∴ The line of symmetry is the perpendicular bisector of AB.
State which of the following figures possess at least one line of symmetry :
(i)
(ii)
(iii)
(iv)
(v)
Answer
(i)
This figure has no line of symmetry. While the outer boundary is a square, the internal path is asymmetrical; folding it vertically, horizontally, or diagonally will not result in matching internal patterns.
(ii)
This figure possesses one line of symmetry. There is a vertical line passing through the centre that divides the figure into identical left and right halves.
(iii)
This figure possesses three lines of symmetry. A line can be drawn through the centre of each "petal" and the midpoint between the opposite two petals.
(iv)
This figure has no line of symmetry. It has rotational symmetry (it looks the same if turned 90° or 180°), but any attempt to fold it will result in the "arms" pointing in opposite directions.
(v)
This figure possesses one line of symmetry. A horizontal line passing through the centres of both circles divides the figure into mirror-image top and bottom halves.
∴ Figures (ii), (iii) and (v) possess at least one line of symmetry.