SYMMETRY (INCLUDING REFLECTION AND ROTATION) - Q&A

1. Multiple Choice Type: Choose the correct answer from the options given below.
(i) The number of line(s) of symmetry of a regular pentagon is:
(a) 4
(b) 5
(c) 0
(d) none of these
Answer: (b) 5
Explanation: A regular polygon with n sides has n lines of symmetry. A regular pentagon has 5 sides, so it has 5 lines of symmetry.

(ii) The number of line(s) of symmetry which a rhombus has :
(a) 4
(b) 2
(c) 0
(d) none of these
Answer: (b) 2
Explanation: A rhombus is symmetrical about its diagonals. Since it has two diagonals, it has 2 lines of symmetry.

(iii) The number of line(s) of symmetry which an isosceles right angled triangle has:
(a) 0
(b) 2
(c) 1
(d) none of these
Answer: (c) 1
Explanation: An isosceles triangle has 1 line of symmetry (the bisector of the vertex angle).

(iv) The number of line(s) of symmetry which a circle of radius 6 cm has :
(a) 0
(b) 2
(c) 1
(d) none of these
Answer: (d) none of these
Explanation: A circle has an infinite number of lines of symmetry (every diameter is a line of symmetry).

(v) The number of line(s) of symmetry of the figure given alongside is:

(a) 1
(b) 2
(c) 4
(d) none of these
Answer: (a) 1
Explanation: The kite shape has one vertical axis of symmetry.

2. State, whether true or false :

(i) The letter B has one line of symmetry.
Answer: True
(The horizontal line passing through the middle.)

(ii) The letter F has no line of symmetry.
Answer: True

(iii) The letter O has only two lines of symmetry.
Answer: True
(Assuming standard typographic 'O' which is oval/elliptical.)

(iv) The figure (parallelogram) has no line of symmetry.
Answer: True

(v) The letter N has one line of symmetry.
Answer: False
(It has rotational symmetry, but no line of symmetry.)

(vi) The figure has one line of symmetry.
Answer: True

(vii) The letter D has only one line of symmetry.
Answer: True

(viii) A scalene triangle has three lines of symmetry.
Answer: False
(A scalene triangle has 0 lines of symmetry.)

3. If possible, draw the largest number of lines of symmetry in each case:

(i) to (vi)
Answer:
(i) Triangle with vertical axis: 1 line.
(ii) Isosceles trapezium: 1 vertical line.
(iii) Triangle: If equilateral, 3 lines. If isosceles, 1 line.
(iv) Shape 'O': 2 lines (vertical and horizontal).
(v) Semi-circle/Shape: 1 vertical line.
(vi) Cylinder/Arrow shape: 1 vertical line.

4. Examine each of the following figures, carefully, and then draw line(s) of symmetry if possible:
(i) to (vi)
Answer:
Draw a dotted line through the center of each figure where the left side mirrors the right side.

5. Draw line(s) of symmetry for each of the following letters:
(i) C
(ii) E
(iii) A
(iv) K
(v) X
(vi) M
(vii) Y
Answer:
(i) C: Horizontal line.
(ii) E: Horizontal line.
(iii) A: Vertical line.
(iv) K: Horizontal line.
(v) X: Vertical and Horizontal lines.
(vi) M: Vertical line.
(vii) Y: Vertical line.

6. Construct a triangle ABC in which AB=AC=5 cm and BC=6 cm. Draw its line(s) of symmetry.
Answer:
Steps:
1. Draw BC = 6 cm.
2. From B and C, draw arcs of 5 cm radius intersecting at A.
3. Join AB and AC.
4. Symmetry: Draw the perpendicular bisector of BC. It passes through A. This is the only line of symmetry.

7. Construct a triangle XYZ in which : XY=YZ=ZX=4.5 cm. Draw all its lines of symmetry.
Answer:
Steps:
1. Draw XY = 4.5 cm.
2. From X and Y, draw arcs of 4.5 cm radius intersecting at Z.
3. Join XZ and YZ to form an equilateral triangle.
4. Symmetry: Draw the angle bisectors of X, Y, and Z. There are 3 lines of symmetry.

8. Construct a triangle PQR in which : PQ=QR=4.2 cm and ∠PQR=90°. Draw all its lines of symmetry.
Answer:
Steps:
1. Draw PQ = 4.2 cm.
2. At Q, construct a 90° angle.
3. Cut off QR = 4.2 cm.
4. Join PR.
5. Symmetry: Draw the bisector of angle Q. It is the only line of symmetry (Isosceles Right Triangle).

9. Mark two points A and B 6.4 cm apart. Construct the line(s) of symmetry so that the points A and B are symmetric with respect to this line.
Answer:
Steps:
1. Draw segment AB = 6.4 cm.
2. Construct the perpendicular bisector of AB.
3. This bisector is the required line of symmetry.

10. Mark two points P and Q 5.3 cm apart. Construct the perpendicular bisector of the line segment PQ. Are the points P and Q symmetric with respect to the perpendicular bisector drawn?
Answer:
Yes, P and Q are symmetric with respect to the perpendicular bisector because they are equidistant from it and lie on a line perpendicular to it.

11. Construct a rectangle ABCD in which AB=7 cm and BC=5 cm. Draw all its line(s) of symmetry.
Answer:
Steps:
1. Construct rectangle ABCD with sides 7 cm and 5 cm.
2. Symmetry: Join the midpoints of opposite sides (AB and CD; AD and BC). There are 2 lines of symmetry.

1. Multiple Choice Type: Choose the correct answer from the options given below.
(i) The point (5, k) after reflection in x-axis becomes (5, -8). The value of k is:
(a) 8
(b) -8
(c) 6
(d) none of these
Answer: (a) 8
Explanation: Reflection in x-axis: (x, y) → (x, -y). Here (5, k) → (5, -k). Since image is (5, -8), -k = -8, so k = 8.

(ii) The point (-k, -6) after reflection in y-axis becomes (-4, -6), the value of k is:
(a) -4
(b) 4
(c) 8
(d) none of these
Answer: (a) -4
Explanation: Reflection in y-axis: (x, y) → (-x, y). Here (-k, -6) → (k, -6). Image is (-4, -6), so k = -4.

(iii) The point (k, -k) after reflection in origin becomes (0, 0); the value of k is:
(a) 2
(b) 1
(c) 1
(d) none of these
Answer: (d) none of these
Explanation: Reflection in origin: (x, y) → (-x, -y). (k, -k) → (-k, k). Since image is (0,0), k=0.

(iv) The point (7, -6) is rotated about origin by 180° in the clockwise direction. The resulting point is:
(a) (-7, -6)
(b) (7, 6)
(c) (-7, 6)
(d) (0, -6)
Answer: (c) (-7, 6)
Explanation: 180° rotation is same as reflection in origin: (x, y) → (-x, -y). (7, -6) → (-7, 6).

(v) The point (-3, 2) is rotated about origin by 90° in the anticlockwise direction. The resulting point is:
(a) (2, 3)
(b) (2, -3)
(c) (-2, -3)
(d) (3, 2)
Answer: (c) (-2, -3)
Explanation: 90° anticlockwise rule: (x, y) → (-y, x). (-3, 2) → (-2, -3).

2. In each of the following cases, write the transformations as required:

(i) Object: (4, -3), Image: (-4, -3)
Answer: Reflection in y-axis
(ii) Object: (-4, 3), Image: (-4, -3)
Answer: Reflection in x-axis
(iii) Object: (-4, -3), Image: (4, 3)
Answer: Reflection in origin (or 180° rotation)
(iv) Object: (0, -7), Image: (0, 7)
Answer: Reflection in x-axis
(v) Object: (8, -5), Image: (-8, 5)
Answer: Reflection in origin
(vi) Object: (-3, 2), Image: (3, -2)
Answer: Reflection in origin
(vii) Object: (5, 8), Image: (-8, 5)
Answer: Rotation 90° anticlockwise
(viii) Object: (-7, 4), Image: (4, 7)
Answer: Rotation 90° clockwise
(ix) Object: (8, 0), Image: (0, -8)
Answer: Rotation 90° clockwise
(x) Object: (3, -2), Image: (-3, 2)
Answer: Reflection in origin

3. Find the co-ordinates of the following points under reflection in x-axis:
(i) (4, 8) → (4, -8)
(ii) (3, -10) → (3, 10)
(iii) (-2, 0) → (-2, 0)

4. Find the reflection of the following points in y-axis:
(i) (9, 10) → (-9, 10)
(ii) (9, 0) → (-9, 0)
(iii) (0, 9) → (0, 9)

5. Find the reflection of the following points in origin:
(i) (5, 4) → (-5, -4)
(ii) (5, -4) → (-5, 4)
(iii) (-5, 4) → (5, -4)
(iv) (-5, -4) → (5, 4)
(v) (0, 4) → (0, -4)

6. Find the co-ordinates of the points obtained on rotating the following points through 180° about the origin:
(i) (3, 4) → (-3, -4)
(ii) (3, -4) → (-3, 4)
(iii) (-3, 4) → (3, -4)
(iv) (-3, -4) → (3, 4)
(v) (0, 4) → (0, -4)

7. Find the co-ordinates of the points obtained on rotating the following points through 90° about origin in the anticlockwise direction:
(i) (4, 6) → (-6, 4)
(ii) (4, -6) → (6, 4)
(iii) (-4, 6) → (-6, -4)
(iv) (-4, -6) → (6, -4)
(v) (0, 6) → (-6, 0)

8. Find the co-ordinates of the points obtained on rotating the following points 90° about origin in the clockwise direction:
(i) (5, 2) → (2, -5)
(ii) (5, -2) → (-2, -5)
(iii) (-5, 2) → (2, 5)
(iv) (-5, -2) → (-2, 5)
(v) (0, 2) → (2, 0)

9. The point P(-3, 12) is reflected in x-axis to point Q. And point Q is then rotated through 180° about origin to point R. Write the co-ordinates of points Q and R.
Answer:
1. P(-3, 12) reflected in x-axis → Q(-3, -12).
2. Q(-3, -12) rotated 180° → R(3, 12).
Q: (-3, -12), R: (3, 12)

10. The point P(-7, 9) is rotated through 90° about origin in the anticlockwise direction to get point Q. If Q is reflected in y-axis to point R, write the co-ordinates of Q and R.
Answer:
1. P(-7, 9) rotated 90° anticlockwise → Q(-9, -7).
2. Q(-9, -7) reflected in y-axis → R(9, -7).
Q: (-9, -7), R: (9, -7)

1. Multiple Choice Type: Choose the correct answer from the options given below.
(i) The number of line(s) of symmetry of a trapezium is:
(a) 0
(b) 1
(c) 2
(d) none of these
Answer: (a) 0

(ii) The number of line(s) of symmetry of an isosceles trapezium is:
(a) 0
(b) 2
(c) 4
(d) none of these
Answer: (d) none of these
(It has 1 line of symmetry.)

(iii) The reflection of point (0, -8) in origin is :
(a) (0, -8)
(b) (0, 8)
(c) (8, 8)
(d) (-8, 0)
Answer: (b) (0, 8)

(iv) The point (-5, -6) is rotated about origin by 90° in the clockwise direction. The resulting point is:
(a) (-6, 5)
(b) (5, 6)
(c) (0, 5)
(d) (6, -5)
Answer: (a) (-6, 5)
Rule (y, -x): (-6, -(-5)) = (-6, 5).

(v) The point (-8, 7) is first reflected in x-axis followed by reflection in origin. The resulting point is:
(a) (-8, -7)
(b) (8, 7)
(c) (8, -7)
(d) (-8, 7)
Answer: (b) (8, 7)
(-8, 7) → x-axis → (-8, -7) → origin → (8, 7).

(vi) Statement 1: A point P is reflected to P' in a line L, then the line L is the right bisector of PP'.
Statement 2: A point P is reflected to P' in a line L, then the line PP' is the right bisector of L.
2. Which of the following options is correct?
(a) Both the statements are true.
(b) Both the statements are false.
(c) Statement 1 is true, and statement 2 is false.
(d) Statement 1 is false, and statement 2 is true.
Answer: (c) Statement 1 is true, and statement 2 is false.

The following questions are Assertion-Reason based questions. Choose your answer based on the codes given below.
(1) Both A and R are correct, and R is the correct explanation for A.
(2) Both A and R are correct, and R is not the correct explanation for A.
(3) A is true, but R is false.
(4) A is false, but R is true.

(vii) Assertion (A): The image of a point A(5, 6) under an anticlockwise rotation of 90° about the origin is A' (-6, 5).
Reason (R): When a point P is rotated through 90° clockwise about the origin, then abscissa of the given point becomes ordinate with opposite sign of the resultant point and the ordinate of the given point becomes the abscissa of the resultant point.
(a) (1)
(b) (2)
(c) (3)
(d) (4)
Answer: (b) (2)
(Both A and R are true statements, but R talks about clockwise while A is anticlockwise, so R doesn't explain A.)

(viii) Assertion (A): A shape (say circle) can be rotated from one position to another about its centre.
Reason (R): A rotation is defined by (i) the angle of rotation (ii) the direction of rotation and the centre of rotation.
(a) (1)
(b) (2)
(c) (3)
(d) (4)
Answer: (a) (1)

(ix) Assertion (A): An equilateral triangle has a point symmetry.
Reason (R): A plane figure is said to have a point symmetry about a point, if every line segment drawn in the given figure passing through it is bisected by this point.
(a) (1)
(b) (2)
(c) (3)
(d) (4)
Answer: (d) (4)
(Equilateral triangle has rotational symmetry of order 3, but not point symmetry. R is the correct definition of point symmetry.)

(x) Assertion (A): A parallelogram has a rotational symmetry of order 2 about the point of intersection O of its diagonals.
Reason (R): A plane figure is said to have rotational symmetry, if it aligns with the original position only once while being rotated through 360°.
(a) (1)
(b) (2)
(c) (3)
(d) (4)
Answer: (c) (3)
(A is true. R is false because aligning "only once" (at 360) means NO rotational symmetry.)

3. Construct a triangle PQR in which : QR=4.6 cm, ∠Q=∠R=50°. Draw all its lines of symmetry.
Answer:
Steps: Draw QR=4.6cm. Draw 50° angles at Q and R to meet at P. Draw the perpendicular bisector of QR. It passes through P and is the only line of symmetry.

4. Construct a triangle ABC in which : AB=BC=4 cm and ∠ABC=60°. Draw all its lines of symmetry.
Answer:
Steps: Draw BC=4cm. At B, draw 60°. Cut AB=4cm. Join AC. Triangle is equilateral. Draw 3 angle bisectors as lines of symmetry.

5. A point (5, -3) is first reflected in x-axis and the resulting point is reflected in origin. Write the co-ordinates of the final point.
Answer: (-5, 3)
(5, -3) → x-axis → (5, 3) → origin → (-5, -3).

6. The point P(-5, 15) is reflected in origin to point Q. And point Q is then rotated through 90° about origin in the clockwise direction to get point R. Write the co-ordinates of points Q and R.
Answer: Q(5, -15), R(-15, -5)
Reflect origin: (-5, 15) → (5, -15). Rotate clockwise 90: (5, -15) → (-15, -5).

7. Name any two geometrical figures, which have two lines of symmetry.
Answer: Rectangle, Rhombus

8. Draw a parallelogram ABCD with AD=BC and ∠B=60°. If possible draw its line(s) of symmetry.
Answer:
A general parallelogram has NO line of symmetry.

9. Draw a kite shaped figure ABCD in which AB=BC=8 cm, AD=CD=5 cm and angle ∠B=60°. Draw its line(s) of symmetry.
Answer:
Draw BD. The line passing through B and D is the only line of symmetry.