Symmetry

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 :

(i) only one line of symmetry each

(ii) two lines of symmetry each

(iii) four lines of symmetry each

(iv) 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 :

(i) one line of symmetry

(ii) two lines of symmetry

(iii) four lines of symmetry

(iv) 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 :

(i) no line of symmetry

(ii) one line of symmetry

(iii) the same number of lines of symmetry as the rhombus

(iv) 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 :

(i) no line of symmetry

(ii) one line of symmetry

(iii) two lines of symmetry

(iv) 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.

Which of the following geometrical figures has exactly one line of symmetry?

(i) A rectangle

(ii) A semicircle

(iii) A regular pentagon

(iv) A rhombus

Answer

A rectangle has 2 lines of symmetry, i.e., horizontal and vertical.

A semicircle has 1 line of symmetry, i.e., the perpendicular bisector of its diameter.

A regular pentagon has 5 lines of symmetry.

A rhombus has 2 lines of symmetry (along its diagonals).

Hence, Option 2 is the correct option.

Which of the following geometrical figures has exactly two lines of symmetry?

(i) A square

(ii) A parallelogram

(iii) An isosceles trapezium

(iv) A rectangle

Answer

A square has 4 lines of symmetry.

A parallelogram has 0 lines of symmetry.

An isosceles trapezium has 1 line of symmetry.

A rectangle has 2 lines of symmetry.

Hence, Option 4 is the correct option.

An equilateral triangle has three lines of symmetry, an isosceles triangle will have:

(i) No line of symmetry

(ii) One line of symmetry

(iii) Two lines of symmetry

(iv) Three lines of symmetry

Answer

An isosceles triangle has two equal sides and two equal angles. It possesses exactly one line of symmetry, which passes through the vertex angle and the midpoint of the base.

Hence, Option 2 is the correct option.

State the type(s) of symmetry possessed by the following figure. Explain each type of symmetry for the figure.

Answer

(1) Linear Symmetry — No linear symmetry.

(2) Point Symmetry — Possesses point symmetry with the centre O as the centre of symmetry.

(3) Rotational Symmetry — Possesses a rotational symmetry of order 4 about the point O.

State the type(s) of symmetry possessed by the following figure. Explain each type of symmetry for the figure.

Answer

(1) Linear Symmetry — No linear symmetry.

(2) Point Symmetry — Possesses point symmetry with the centre O as the centre of symmetry.

(3) Rotational Symmetry — Possesses a rotational symmetry of order 2 about the point O.

State the type(s) of symmetry possessed by the following figure. Explain each type of symmetry for the figure.

Answer

(1) Linear Symmetry — One line of symmetry, i.e., the vertical line passing through vertices A and C.

(2) Point Symmetry — No point symmetry.

(3) Rotational Symmetry — No rotational symmetry.

State the type(s) of symmetry possessed by the following figure. Explain each type of symmetry for the figure.

Answer

(1) Linear Symmetry — Three lines of symmetry, i.e., AD, BE and CF.

(2) Point Symmetry — No point symmetry.

(3) Rotational Symmetry — Possesses rotational symmetry of order 3 about the centre D.

State the type(s) of symmetry possessed by the following figure. Explain each type of symmetry for the figure.

Answer

(1) Linear Symmetry — One line of symmetry, i.e., the line parallel to AB passing through the point C.

(2) Point Symmetry — No point symmetry.

(3) Rotational Symmetry — No rotational symmetry.

State the type(s) of symmetry possessed by the following figure. Explain each type of symmetry for the figure.

Answer

(1) Linear Symmetry — Two lines of symmetry, i.e., one vertical and one horizontal line passing through the centre O.

(2) Point Symmetry — Possesses point symmetry with the centre O as the centre of symmetry.

(3) Rotational Symmetry — Possesses rotational symmetry of order 2 about the centre O.

State the type(s) of symmetry possessed by the following figure. Explain each type of symmetry for the figure.

Answer

(1) Linear Symmetry — One line of symmetry, i.e., the bisector of the angle AOB.

(2) Point Symmetry — No point symmetry.

(3) Rotational Symmetry — No rotational symmetry.

State the type(s) of symmetry possessed by the following figure. Explain each type of symmetry for the figure.

Answer

(1) Linear Symmetry — No linear symmetry.

(2) Point Symmetry — Possesses point symmetry with the centre O as the centre of symmetry.

(3) Rotational Symmetry — Possesses a rotational symmetry of order 2 about the point O.

State the type(s) of symmetry possessed by the following figure. Explain each type of symmetry for the figure.

Answer

(1) Linear Symmetry — Two lines of symmetry, i.e., (i) the line joining points A and C and (ii) the line joining points B and D.

(2) Point Symmetry — The point O of intersection of the lines AC and BD is the centre of symmetry.

(3) Rotational Symmetry — Possesses a rotational symmetry of order 2 about the point O.

State the type(s) of symmetry possessed by the following figure. Explain each type of symmetry for the figure.

Answer

(1) Linear Symmetry — No linear symmetry.

(2) Point Symmetry — No point symmetry.

(3) Rotational Symmetry — Possesses rotational symmetry of order 3 about the centre O.

A line segment is symmetrical about

1. any line perpendicular to it.
2. any line passing through its mid-point
3. any line parallel to it.
4. its perpendicular bisector.

Answer

A line segment has two lines of symmetry: itself and its perpendicular bisector. Among the options, only the perpendicular bisector is listed.

Hence, Option 4 is the correct option.

A scalene triangle has

1. no line of symmetry
2. one line of symmetry
3. two lines of symmetry
4. three lines of symmetry

Answer

In a scalene triangle, all sides and angles are unequal, so there is no line that can divide it into two identical mirror images.

Hence, Option 1 is the correct option.

The lines of symmetry of a rhombus are

1. perpendicular bisector of each of its sides
2. its two diagonals
3. the lines joining the mid points of its opposite sides
4. its sides

Answer

Folding a rhombus along its diagonals results in the opposite halves matching perfectly.

Hence, Option 2 is the correct option.

A circle has

1. no line of symmetry
2. one line of symmetry
3. two lines of symmetry
4. an unlimited number of lines of symmetry

Answer

Any straight line passing through the centre of a circle (a diameter) is a line of symmetry.

Hence, Option 4 is the correct option.

The line of symmetry of a rectangle are

1. its four sides
2. its two diagonals
3. the bisectors of its four interior angles
4. the lines joining the midpoints of its opposite sides.

Answer

A rectangle is symmetrical about the vertical and horizontal lines passing through the midpoints of its sides.

Hence, Option 4 is the correct option.

ABCD is a kite in which AB = AD and CB = CD. The kite is symmetrical about

1. the diagonal AC
2. the diagonal BD
3. both the diagonals AC and BD
4. the lines joining the mid-points of its opposite sides.

Answer

In a kite, the diagonal connecting the vertices where equal sides meet (AC) acts as the axis of symmetry.

Hence, Option 1 is the correct option.

In △ABC, AB = AC and AD ⊥ BC, BE ⊥ CA and CF ⊥ AB. Then, △ABC is symmetrical about

1. AD
2. BE
3. CF
4. BC

Answer

Since the triangle is isosceles (AB = AC), it has one line of symmetry which is the altitude drawn from the vertex angle to the base (AD).

Hence, Option 1 is the correct option.

Which amongst the following letters has the highest number of lines of symmetry?

1. A
2. K
3. X
4. N

Answer

'A' has 1, 'K' has 1, 'N' has 0, and 'X' has 2 (horizontal and vertical) lines of symmetry.

Hence, Option 3 is the correct option.

Which of the following letters of the English alphabet does not possess a point symmetry?

1. C
2. N
3. S
4. X

Answer

'N', 'S', and 'X' look the same when rotated through 180°. 'C' does not.

Hence, Option 1 is the correct option.

An equilateral triangle has a rotational symmetry of the order

1. 1
2. 2
3. 3
4. 4

Answer

An equilateral triangle maps onto itself at 120°, 240° and 360° rotations.

Hence, Option 3 is the correct option.

An isosceles triangle possesses

1. linear symmetry
2. point symmetry
3. rotational symmetry
4. all of these

Answer

It has one line of symmetry but does not look the same if rotated or turned upside down.

Hence, Option 1 is the correct option.

A regular pentagon does not possess

1. linear symmetry
2. point symmetry
3. rotational symmetry
4. all of these

Answer

A regular pentagon has an odd number of sides (5). In regular polygons, point symmetry (which is the same as rotational symmetry of order 2) only exists if the number of sides is even.

Hence, Option 2 is the correct option.

A parallelogram does not possess

1. linear symmetry
2. point symmetry
3. rotational symmetry
4. all of these

Answer

A general parallelogram cannot be folded to create mirror images, though it does have point and rotational symmetry.

Hence, Option 1 is the correct option.

An equilateral triangle does not possess

1. linear symmetry
2. point symmetry
3. rotational symmetry
4. none of these

Answer

Rotating an equilateral triangle $180^\circ$ leaves it pointing downward, so it does not match its original position.

Hence, Option 2 is the correct option.

Which of the following letters of English alphabet has a rotational symmetry?

1. C
2. K
3. N
4. T

Answer

'N' has rotational symmetry of order 2 (180° rotation). 'C', 'K', and 'T' only look the same after a full 360° turn.

Hence, Option 3 is the correct option.

Fill in the blanks :

(i) A circle has ............... lines of symmetry.

(ii) The letter S does not possess ............... symmetry.

(iii) A semi-circle is symmetrical about the ............... of its diameter.

(iv) The letter H has ............... line(s) of symmetry.

(v) A quadrilateral having 4 lines of symmetry as well as rotational symmetry of order 4 is ............... .

Answer

(i) A circle has infinite lines of symmetry.

(ii) The letter S does not posses linear symmetry.

(iii) A semi-circle is symmetrical about the perpendicular bisector of its diameter.

(iv) The letter H has two line(s) of symmetry.

(v) A quadrilateral having 4 lines of symmetry as well as rotational symmetry of order 4 is square.

Explanation

(i) Every diameter of a circle acts as a fold line that splits it into two identical halves. Since we can draw infinitely many diameters, there are infinitely many lines of symmetry.

(ii) If we try to fold the letter S vertically or horizontally, the curves point in opposite directions. It only looks the same if we rotate it 180° (rotational symmetry).

(iii) For a semicircle, the only way to get matching halves is to fold it exactly down the middle of the flat edge (the diameter).

(iv) The letter H is balanced both left-to-right and top-to-bottom, giving it two axes of reflection.

(v) While a rectangle has rotational symmetry of order 2 and two lines of symmetry, only a square reaches "perfection" with 4 lines (including diagonals) and a matching rotational order of 4.

Write true (T) or false (F) :

(i) A kite possesses a linear symmetry but no rotational symmetry.

(ii) The order of rotational symmetry of a regular hexagon is 6.

(iii) A parallelogram does not have any line of symmetry.

(iv) A square has a point symmetry but rhombus does not.

(v) The letter N does not possess a rotational symmetry.

Answer

(i) True
Reason — A kite has exactly one line of symmetry i.e., the diagonal connecting the vertices of the equal sides. However, it does not look like its original self at any point during a rotation until it completes a full 360° turn.

(ii) True
Reason — For any regular polygon, the order of rotational symmetry is equal to the number of its sides. Since a regular hexagon has 6 equal sides and angles, it maps onto itself 6 times in one full rotation.

(iii) True
Reason — A general parallelogram cannot be folded along any line to produce two matching halves. While it has rotational symmetry, it lacks linear symmetry.

(iv) False
Reason — Both a square and a rhombus possess point symmetry. Any figure that looks the same after a 180° rotation (upside down) has point symmetry. Since both shapes map onto themselves after a half-turn, they both have it.

(v) False
Reason — The letter N possesses rotational symmetry of order 2. If we rotate the letter N by 180°, it looks exactly the same as it did in its starting position.

Assertion: Order of rotational symmetry for the given figure is 4.

Reason: A figure is said to possess rotational symmetry if it fits on itself more than once while being rotated through 360°.

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
2. Both Assertion (A) and Reason (R) are true and Reason (R) is not the correct explanation of Assertion (A).
3. Assertion (A) is true but Reason (R) is false.
4. Assertion (A) is false but Reason (R) is true.

Answer

Assertion (A) is false but Reason (R) is true.

Explanation

The given figure consists of two circles touching each other. To look exactly the same, you would need to rotate it by 180° (half-turn) or 360° (full turn). This means the figure fits on itself twice in one full rotation. Therefore, the order of rotational symmetry is 2, not 4.

So, Assertion is false.

The statement given in the reason is correct and is the standard mathematical definition of rotational symmetry.

So, Reason is true.

Hence, option 4 is the correct option.

Assertion: The number of lines of symmetry of a regular polygon is equal to its number of vertices.

Reason: A figure that possesses point symmetry always has line symmetry.

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
2. Both Assertion (A) and Reason (R) are true and Reason (R) is not the correct explanation of Assertion (A).
3. Assertion (A) is true but Reason (R) is false.
4. Assertion (A) is false but Reason (R) is true.

Answer

Assertion (A) is true but Reason (R) is false.

Explanation

For any regular polygon (like an equilateral triangle, square, or regular pentagon), the number of lines of symmetry is equal to the number of sides, which is also equal to the number of vertices.

For example, a square has 4 vertices and 4 lines of symmetry.

So, Assertion is true.

Point symmetry is rotational symmetry of order 2 (looking the same upside down). However, a figure can have point symmetry without having any lines of symmetry.

A classic example is the letter 'S' or a parallelogram. You can rotate them 180° to match, but you cannot fold them to get mirror images.

So, Reason is false.

Hence, option 3 is the correct option.