ML Aggarwal Class 8 Maths Solutions Chapter 16 Visualising Solid Shapes

Exercise 16

Question 1. Draw the line or lines of symmetry, if any, of the following shapes and count their number:
(i) Two overlapping circles
(ii) A circle with a smaller circle inside it
(iii) A spiral or curved shape
(iv) A plus sign or cross
(v) A three-petaled flower shape
(vi) A four-petaled flower shape
Answer: Based on the figures shown:
(i) One line of symmetry (horizontal line through the centres)
(ii) One line of symmetry (vertical line through the centres)
(iii) No line of symmetry
(iv) Four lines of symmetry (two diagonal lines and two perpendicular lines)
(v) Three lines of symmetry (one through each petal)
(vi) Four lines of symmetry (one through each petal)
In simple words: A line of symmetry is an imaginary line that divides a shape into two matching halves. When you fold the shape along this line, both sides fit perfectly on top of each other.

Exam Tip: Count all possible lines that divide a shape into mirror images - check both vertical and diagonal directions, not just horizontal ones.

Question 2. For each of the given shape in question 1, find the order of the rotational symmetry (if any).
Answer:
(i) No rotational symmetry
(ii) No rotational symmetry
(iii) Rotational symmetry of order 2
(iv) Rotational symmetry of order 4
(v) Rotational symmetry of order 3
(vi) Rotational symmetry of order 4
In simple words: Rotational symmetry means a shape looks the same when you spin it around a central point. The order tells you how many times it matches itself in a complete 360-degree turn.

Exam Tip: To find the order, divide 360 degrees by the angle you need to rotate the shape for it to look identical - this gives you the rotational symmetry order.

Question 3. Construct a rectangle ABCD such that AB = 4.5 cm and BC = 3 cm. Draw its line (or lines) of symmetry.
Answer:
Steps to construct:
Step 1: Draw a line segment AB measuring 4.5 cm.
Step 2: From point B, draw a line BQ that makes a 90-degree angle with AB.
Step 3: Using B as your centre point and a radius of 3 cm, mark a point C on line BQ.
Step 4: With C as the centre and radius 4.5 cm, draw an arc. Also, with A as the centre and radius 3 cm, draw another arc that crosses the first arc - this crossing point is D.
Step 5: Join points C and D, and also join points A and D.
Step 6: ABCD is now your required rectangle.
Lines of symmetry: The rectangle has 2 lines of symmetry. These are the lines that join the midpoints of the opposite sides (one vertical line through the middle and one horizontal line through the middle).
In simple words: A rectangle can be folded in half two ways - once down the middle vertically and once across the middle horizontally - and both halves match exactly.

Exam Tip: Always draw the construction lines lightly with a pencil, and mark the lines of symmetry with a dotted line to show they are not part of the figure itself.

Question 4. Construct a rhombus ABCD with AB = 5.3 cm and ∠A = 60°. Draw its line (or lines) of symmetry.
Answer:
Steps to construct:
Step 1: Draw a line segment AB measuring 5.3 cm.
Step 2: At point A, construct an angle of 60 degrees.
Step 3: Using A as your centre and a radius of 5.3 cm, draw an arc along the angle line you just made. Mark the point where the arc meets the line as D.
Step 4: With D as the centre and radius 5.3 cm, draw another arc. Also, with B as the centre and the same radius (5.3 cm), draw an arc that crosses the first arc - mark this crossing point as C.
Step 5: Join points C and D, and also join points B and C.
Step 6: ABCD is now your required rhombus.
Lines of symmetry: A rhombus has 2 lines of symmetry. These are the two diagonals of the rhombus - they cross each other at the centre and divide the rhombus into two equal mirror-image parts each.
In simple words: If you draw both diagonals of a rhombus (the lines connecting opposite corners), each one acts as a mirror line - the shape reflects perfectly on both sides of each diagonal.

Exam Tip: The diagonals of a rhombus are always perpendicular (at 90 degrees) to each other - use this fact to check your construction is correct.

Objective Type Questions

Question 5. Fill in the blanks:
(i) A figure has __________ symmetry if it is its own image under a reflection.
(ii) A kite has __________ line(s) of symmetry.
(iii) A parallelogram has __________ line(s) of symmetry.
(iv) The centre of rotation of an equilateral triangle is the point of intersection of its __________.
(v) The centre of rotation of a rhombus is the point __________.
(vi) A regular polygon of n-sides has __________ number of lines of symmetry.
(vii) Angle of rotational symmetry in an equilateral triangle is __________.
(viii) Angle of rotational symmetry in a regular pentagon is __________.
(ix) If after a rotation of 45° about a fixed point the figure looks exactly the same, then the order of rotational symmetry is __________.
Answer:
(i) line
(ii) one
(iii) none
(iv) angle bisectors/altitudes/medians
(v) of intersection of its diagonals
(vi) n
(vii) 120°
(viii) 72°
(ix) 8
In simple words: Line symmetry means you can fold a shape and both sides match. The number of lines of symmetry varies by shape - squares have more than rectangles, and regular shapes have symmetry lines equal to their number of sides.

Exam Tip: Remember that rotational symmetry order = 360° divided by the rotation angle - for 45°, that gives 360 ÷ 45 = 8.

Question 6. State whether the following statements are true (T) or false (F):
(i) A parallelogram has diagonals as its lines of symmetry.
(ii) A regular triangle has three lines of symmetry, one point of symmetry and has rotational symmetry of order 3.
(iii) A regular quadrilateral has four lines of symmetry, one point of symmetry and has rotational symmetry of order 4.
(iv) A parallelogram has no rotational symmetry.
(v) A regular pentagon has one point of symmetry.
(vi) The letter Z has one line of symmetry.
Answer:
(i) False. A parallelogram does not have any line of symmetry.
(ii) False. An equilateral triangle has three lines of symmetry (one through each vertex), one point of symmetry, and rotational symmetry of order 3. However, a square (regular quadrilateral) has these properties, not a regular triangle in the way the statement describes. A rectangle has 2 lines of symmetry, one point of symmetry, and rotational symmetry of order 2.
(iii) True. A square (regular quadrilateral) has four lines of symmetry (two through opposite vertices and two through midpoints of opposite sides), one point of symmetry at its centre, and rotational symmetry of order 4.
(iv) False. A parallelogram has rotational symmetry of order 2 - when rotated 180 degrees, it looks the same.
(v) False. A regular pentagon has no point of symmetry because it has an odd number of sides.
(vi) False. The letter Z has no line of symmetry - if you fold it in any direction, the two halves do not match.
In simple words: Line symmetry occurs when a shape can be folded so both sides match. Point symmetry means a shape looks the same when rotated 180 degrees around a central point. Not all shapes have both types.

Exam Tip: For regular polygons, the number of lines of symmetry equals the number of sides only if the shape is truly regular - check that all sides and angles are equal before deciding.

Question 3. The number of lines of symmetry which a quadrilateral cannot have
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (c) 3
In simple words: A quadrilateral (four-sided shape) can have 0, 1, 2, or 4 lines of symmetry, but never exactly 3 lines. For instance, a kite has 1 line, a rectangle has 2 lines, and a square has 4 lines.

Exam Tip: Test each quadrilateral type - kite (1), rectangle (2), rhombus (2), square (4), trapezoid (0 or 1) - to confirm that 3 is impossible.

Question 4. A possible angle of rotation of a figure having rotational symmetry of order greater than or equal to 2 is
(a) 36°
(b) 144°
(c) 150°
(d) 360°
Answer: (a) 36°
In simple words: If a shape has rotational symmetry of order 2 or higher, the rotation angle must divide evenly into 360 degrees. Among these options, 36° divides into 360 exactly 10 times, so it works. The number 144° does not divide evenly (360 ÷ 144 is not a whole number), and 150° does the same, while 360° is a complete turn.

Exam Tip: Use the formula: rotation angle = 360° ÷ (order of rotational symmetry). For order 2 or greater, the angle must be a factor of 360.

Question 5. The figure which does not have both line and rotational symmetry is
(a) A plus or cross shape
(b) A circle
(c) An irregular angle or bent shape
(d) A three-petaled flower or propeller
Answer: (c) An irregular angle or bent shape
In simple words: Most regular shapes have both kinds of symmetry, but an uneven bent or angled figure (option c) typically has neither. A cross, circle, and three-petal shape all have both line symmetry and rotational symmetry, but a random bent shape does not.

Exam Tip: Look for shapes that break regular patterns - if a shape is lopsided or asymmetrical, it is unlikely to have either form of symmetry.

Question 6. The letter which has both line and rotational symmetry is
(a) H
(b) M
(c) S
(d) Y
Answer: (a) H
In simple words: The letter H has a vertical line down the middle that divides it into two matching halves (line symmetry). It also looks the same when rotated 180 degrees (rotational symmetry of order 2). The letters M, S, and Y do not have both properties.

Exam Tip: Check capital letters for vertical and horizontal lines of symmetry first, then rotate them 180 degrees mentally to confirm rotational symmetry - H, I, N, O, S, X, and Z are worth testing.