Symmetry, Reflection and Rotation — Statement I-II Type Questions

Statement I: The diagonals of a rhombus are its mirror lines.

Statement II: A rhombus has a rotational symmetry of order 2.

1. Statement I is true but Statement II is false.

2. Statement I is false but Statement II is true.

3. Both Statement I and Statement II are true.

4. Both Statement I and Statement II are false.

Answer

A rhombus has two lines of symmetry, which are its diagonals. On folding the rhombus about either of its diagonals, the two parts coincide. So, the diagonals of a rhombus are its mirror lines.

So, Statement I is true.

On rotating a rhombus through 180° about the point of intersection of its diagonals, it looks exactly the same as in its starting position.

Order of rotational symmetry = $\dfrac{360\degree}{180\degree} = 2$.

So, Statement II is true.

∴ Both Statement I and Statement II are true.

Hence, option 3 is the correct option.

Statement I: A regular pentagon has one point of symmetry.

Statement II: A regular hexagon has one point of symmetry.

1. Statement I is true but Statement II is false.

2. Statement I is false but Statement II is true.

3. Both Statement I and Statement II are true.

4. Both Statement I and Statement II are false.

Answer

A figure has a point of symmetry if it looks exactly the same on being rotated through 180° about that point.

On rotating a regular pentagon through 180° about its centre, it does not look the same as in its starting position. So, a regular pentagon has no point of symmetry.

So, Statement I is false.

On rotating a regular hexagon through 180° about its centre, it looks exactly the same as in its starting position. So, a regular hexagon has one point of symmetry, which is its centre.

So, Statement II is true.

∴ Statement I is false but Statement II is true.

Hence, option 2 is the correct option.

Statement I: In the adjoining figure, the dotted line is the line of symmetry.

Statement II: A plane figure is symmetrical about a line if it is divided into two coincident parts by that line.

1. Statement I is true but Statement II is false.

2. Statement I is false but Statement II is true.

3. Both Statement I and Statement II are true.

4. Both Statement I and Statement II are false.

Answer

The figure given is the letter S. On folding the letter S about the dotted line, the two parts do not coincide. In fact, the letter S has no line of symmetry but it has a point of symmetry.

So, Statement I is false.

A plane figure is symmetrical about a line if that line divides it into two coincident parts, and such a line is called its line of symmetry. This is the definition of line symmetry.

So, Statement II is true.

∴ Statement I is false but Statement II is true.

Hence, option 2 is the correct option.

Statement I: The angle of rotational symmetry in a regular hexagon is 30°

Statement II: If A° (≤ 180°) is the smallest angle through which a figure can be rotated and still looks the same, then it has a rotational symmetry of order $\dfrac{180\degree}{A\degree}$

1. Statement I is true but Statement II is false.

2. Statement I is false but Statement II is true.

3. Both Statement I and Statement II are true.

4. Both Statement I and Statement II are false.

Answer

A regular hexagon has rotational symmetry of order 6.

∴ Angle of rotational symmetry = $\dfrac{360\degree}{6} = 60\degree$, and not 30°.

So, Statement I is false.

If A° (≤ 180°) is the smallest angle through which a figure can be rotated and still looks the same, then it has a rotational symmetry of order $\dfrac{360\degree}{A\degree}$ and not $\dfrac{180\degree}{A\degree}$.

So, Statement II is false.

∴ Both Statement I and Statement II are false.

Hence, option 4 is the correct option.