case /data -based (4 Marks)

Question 14 Marks

Here is a picture of a car wheel.

What is the order of rotational symmetry for this wheel?

Answer

Here, in this case there are four positions in which the trim will look exactly the same. These occur when the wheel is rotated through one-fourth of a complete turn, two-fourth and three-fourth of a turn and finally a complete turn (when, of course, the wheel is back in its original position).
Therefore, the order of rotational symmetry of wheel is 4.

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Question 24 Marks

A quadrilateral is drawn on a square grid. $O$ is the dot marked on one vertex of the quadrilateral.

Q. 1. How many lines of symmetry are there in this quadrilateral?
(a) 0 $\quad$ (b) 1 $\quad$ (c) 2 $\quad$ (d) 4
Q. 2. Draw three more congruent quadrilaterals around 0 so that the complete figure has rotational symmetry of order four.
Q. 3. Which of the following statements about a parallelogram's symmetry is true?
(a) A parallelogram has neither a line of symmetry nor rotational symmetry,
(b) A parallelogram has a line of symmetry but no rotational symmetry.
(c) A parallelogram has a point symmetry and rotational symmetry but no line of symmetry.
(d) A parallelogram has rotational symmetry but no point or line symmetry.
Q. 4. Jyoti claims that rotational symmetry of order 1 implies no rotational symmetry. Do you agree or disagree with her claim? Give reasons.

Answer

1. (a) A line of symmetry is the line that divides a shape or an object into two equal and symmetrical parts and also called as axis of symmetry.
Here, we can see that there are no line of symmetry Hence, the required answer is 0.
2. By drawing three more congruent quadrilaterals from the reference point O , the figure will be

So, the complete figure has rotational symmetry of order four.
Hence, the required answer is obtained.
3. (c) A parallelogram has a point of symmetry and rotational symmetry but no line symmetry.
4. Agree.
If the given angle is a factor of $360^{\circ}$, only then the figure will have rotational symmetry of order more than one.
For example
$45^{\circ}$ is a factor of $360^{\circ}$, so the figure will have rotational symmetry of order more than 1 and there would be 8 rotations.

Hence, every objects looks similar after $360^{\circ}$ rotation.

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Question 34 Marks

The picture shows a girl and her reflection in a mirror.

Q. 1. Can you draw a line of symmetry on this picture? Mention Yes or No. Justify your choice.
Q. 2. The girl has her right hand raised. Why does it look like her left hand in the mirror image?
Q. 3. Points, $P, Q, R$ and $S$ are marked on the girl and their mirror reflections $P^{\prime}, Q^{\prime}, R^{\prime}$ and $S'$ are marked on the image. Which point and its image in the mirror have the greatest distance between them?
(a) $P$ and $Q$
(b) $Q$ and $Q^{\prime}$
(c) $R$ and $R^{\prime}$
(d) $P^{\prime}$ and $S^{\prime}$
Q. 4. Which type of symmetry does the picture show?
(a) Line symmetry
(b) Point symmetry
(c) Rotation symmetry
(d) Reflection symmetry

Answer

1. A line of symmetry is the line that divides a shape of an object into equal and symmetrical parts. We call this line axis of symmetry or mirror line because it divides the figure symmetrically and the divided parts look like mirror reflections of each other.
Here, they have already gave the reflection and there is no fixed line which divides the figure into two identical parts.
Hence, drawing a line of symmetry is not possible.
2. A mirror image (i.e. in a plane mirror) is a reflected duplication of an object that appears almost identical but is reversed in the direction perpendicular to the mirror surface.
So, in mirror reflection, right-orientation changes to left orientation.
That is why, when the girl has her right hand raised, it looks like her left hand in the mirror image.
3. (c) We have to find the longest distance of points given i.e. $P, Q, R, S$ and $P^{\prime}, Q^{\prime}, R^{\prime}$ and $S^{\prime}$.
$P$ is the real image and $P^{\prime}$ is the image in the mirror reflection.
Since, we know that a mirror image is a reflected duplication of an object that appears almost identical but in reversed in the direction perpendicular to the mirror surface.
We can see that in figure, the distance between $R$ and $R^{\prime}$ is the longest or the greatest.
4. (d) The given image is same as image in mirror, so the symmetry is reflection symmetry
Hence, the required answer is reflection symmetry.

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MATHS case /data -based (4 Marks)

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