NCERT MCQ Solutions for Class 6 Maths Chapter 9 Symmetry

A line of symmetry makes the two parts mirror halves. Folding along the line places one half exactly over the other. Equal areas alone are not enough, because their shapes and positions must match too.

A square has a vertical and a horizontal line through the midpoints of opposite sides. Its two diagonals are also lines of symmetry.
Total lines of symmetry = 2 + 2 = 4

The two lines through the midpoints of opposite sides divide the rectangle into matching mirror halves. Its diagonals do not give matching halves when folded.
Number of lines of symmetry = 2

The diagonal divides the rectangle into equal-area triangles, but folding along it does not make them overlap exactly. The longer side does not match the shorter side. Therefore, the diagonal is not a line of symmetry.

Each line from a vertex to the midpoint of the opposite side divides this triangle into mirror halves. There are three vertices, giving three such lines.
Number of lines of symmetry = 3

If a shape has both reflection and rotational symmetry it has both types of symmetry.

A regular pentagon has 5 lines of symmetry.

A triangle may have no line of symmetry, one line, or three lines. If two different symmetry lines exist, the matching conditions force all three sides to be equal, which gives a third line too. Exactly two is impossible.

A symmetry fold in a triangle would match two sides with equal lengths. Here all three side lengths are different, so no such match is possible. The triangle therefore has no line of symmetry.

Three symmetry lines join pairs of opposite vertices, and three pass through the midpoints of opposite sides. Each folds the regular hexagon into matching halves.
Total = 3 + 3 = 6 lines of symmetry

The vertical symmetry line exchanges the left and right halves while keeping the height unchanged. Therefore, the top-left corner A matches the top-right corner B. Similarly, the bottom corners D and C exchange positions.

Diagonal AC is the fold line, so A and C stay fixed. The other two corners lie on opposite sides of this diagonal and match when folded. Thus, B moves to D and D moves to B.

A scalene triangle has no line of symmetry.

Symmetry is a pattern that repeats in a definite way.

A point on the reflection line has no distance from that line. Folding or reflecting leaves it in place. For example, the corners on a square’s diagonal remain fixed when that diagonal is used as the symmetry line.

Pressing the halves together transfers each paint mark to a matching position on the other half. Opening the sheet produces mirror images on the two sides. The original crease is therefore a line of symmetry of the print.

The punch passes through two layers, making one hole in each. When the sheet opens, the holes lie on opposite sides of the fold at matching positions.
Total holes = 2
The fold is their line of symmetry.

The two folds create four layers. A single punch makes a hole in each layer, and the holes separate when both folds are opened.
Total holes = 4
The resulting pattern is symmetric about both original fold lines.

Mirror points lie at equal distances on opposite sides of the mirror line. For a vertical mirror line, their horizontal row stays the same.
Left distance = right distance = 2 grid spaces

Smallest angle of rotational symmetry for a regular hexagon is 120°.

A circle has infinite lines of symmetry.

During rotation, the figure turns about one fixed point. This point is the centre of rotation. Other points move around it, and a successful symmetry rotation brings the whole figure exactly onto its original position.

A full turn brings every point back to its original position.
Full turn = 360°
Therefore, 360° is always an angle of symmetry. Smaller rotations work only when the figure has the required repeating pattern.

Rotational symmetry requires a matching turn greater than 0° and less than 360°. This figure has no such turn. Its match after 360° is simply the return to the original position that occurs for every figure.

The matching rotations in one full turn are 90°, 180°, 270° and 360°.
Order = 360° ÷ 90° = 4
The full-turn position is counted once, while the starting position at 0° is not counted again.

A 180° rotation is half a turn. It carries each corner of the square to its opposite corner. Since C is opposite A, A moves to C; similarly, B moves to D.

A square have 4 lines of symmetry.

The order of rotational symmetry of an equilateral triangle is 3.

The three equal gaps make one full turn.
Angle between neighbouring arms = 360° ÷ 3 = 120°
With identical arms, rotations of 120°, 240° and 360° bring the design onto itself.

The five identical arms divide a full turn into five equal steps. One step carries each arm to the next.
Smallest angle = 360° ÷ 5 = 72°
Five such steps complete the full turn.

The full turn is divided into seven equal steps.
Smallest angle = 360° ÷ 7
= 51 3/7°
Since 7 × 51 = 357, the remaining 3 degrees give the fractional part 3/7°.

The matching angles are multiples of the smallest angle, 60°.
2 × 60° = 120°, 3 × 60° = 180°, 4 × 60° = 240°
5 × 60° = 300°, 6 × 60° = 360°
These are all the remaining matches within one full turn.

Matching angles occur in equal steps from the smallest one. With exactly two positive matches before 60°, the first three are one step, two steps and three steps.
One step = 60° ÷ 3 = 20°
The two earlier angles are 20° and 40°.

Repeated equal symmetry turns must complete a full 360° turn.
17 × 21 = 357, which is not 360.
So 360 ÷ 17 is not a whole number, and 17° cannot be the smallest positive symmetry angle.

Each matching turn advances by 45°. The number of these steps in one full turn is:
Order = 360° ÷ 45° = 8
Thus, the figure has eight matching positions from 45° through 360°.

All points of a circle are at the same distance from its centre. Turning the circle about that centre keeps this distance unchanged and leaves the same circular boundary. Therefore, every rotation angle gives an exact match.

Every line through the centre divides the circle into matching mirror halves. Each diameter lies along such a line. Since diameters can be drawn in infinitely many directions, the circle has infinitely many lines of symmetry.

The angle from one spoke to the next is 360° ÷ 24 = 15°. Each 15° step moves every spoke to the next matching position.
Order = 360° ÷ 15° = 24
There are 24 matching rotations up to and including 360°.

A line that divides a figure into two equal mirror halves is called axis of symmetry.

In rotational symmetry the centre of rotation is called centre of rotation.