Exploring Some Geometric Themes — Class 8 Mathematics NCERT Solutions (Free)
Free step-by-step NCERT solutions for Class 8 Mathematics chapter "Exploring Some Geometric Themes" — 8 important questions with detailed answers for CBSE board exam preparation.
TL;DR: Free step-by-step NCERT solutions for Class 8 Mathematics chapter "Exploring Some Geometric Themes" — 8 important questions with detailed answers for…
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NCERT Solutions for Class 8 Mathematics — Exploring Some Geometric Themes. Step-by-step answers to all 8 textbook questions from this chapter, written for CBSE board preparation and free to use.
Exploring Some Geometric Themes — All 8 Questions Solved
Q1. Construct a line segment AB of length 5 cm. What is the procedure?
Step 1: Place the ruler on paper.
Step 2: Mark point A at 0 cm mark on the ruler.
Step 3: Mark point B at 5 cm mark on the ruler.
Step 4: Use a pencil to draw a straight line connecting A and B.
Step 5: AB = 5 cm (the required line segment)
Q2. How do you construct the perpendicular bisector of a line segment?
Step 1: Given line segment AB.
Step 2: Set compass to more than half the length of AB.
Step 3: Draw arcs with center A, one above and one below the line.
Step 4: With the same compass width, draw arcs with center B, intersecting the previous arcs.
Step 5: Label intersection points as P and Q.
Step 6: Draw line PQ—this is the perpendicular bisector of AB.
Q3. Construct an angle of 60° using compass and straightedge.
Step 1: Draw a line and mark point O on it.
Step 2: With O as center, draw an arc of any radius intersecting the line at point A.
Step 3: With the same radius and A as center, draw an arc intersecting the first arc at point B.
Step 4: Draw a straight line from O through B.
Step 5: Angle AOB = 60°
Q4. What is the order of rotational symmetry of a square?
Step 1: Rotational symmetry means the shape looks the same after rotation.
Step 2: A square looks identical after rotation by 90°, 180°, 270°, and 360°.
Step 3: The shape matches its original position 4 times in a complete 360° rotation.
Step 4: Order of rotational symmetry = 4
Q5. How many lines of symmetry does a rectangle have?
Step 1: A line of symmetry divides a shape into two identical halves.
Step 2: A rectangle has 1 horizontal line of symmetry (through the middle length).
Step 3: A rectangle has 1 vertical line of symmetry (through the middle width).
Step 4: The diagonals of a rectangle are NOT lines of symmetry.
Step 5: Total lines of symmetry = 2
Q6. Construct a triangle ABC with sides AB = 4 cm, BC = 5 cm, and AC = 6 cm.
Step 1: Draw a line and mark point A.
Step 2: Using ruler, mark point B at 4 cm from A. (AB = 4 cm)
Step 3: With compass set to 5 cm, center at B, draw an arc above the line.
Step 4: With compass set to 6 cm, center at A, draw an arc intersecting the previous arc. Label this point C.
Step 5: Draw lines AC and BC.
Step 6: Triangle ABC is constructed with the given side lengths.
Q7. What is the order of rotational symmetry of an equilateral triangle?
Step 1: An equilateral triangle has all sides and angles equal.
Step 2: It looks identical after rotation by 120°, 240°, and 360°.
Step 3: The shape matches its original position 3 times in a complete rotation.
Step 4: Order of rotational symmetry = 3
Q8. How many lines of symmetry does a regular hexagon have?
Step 1: A regular hexagon has 6 equal sides and 6 equal angles.
Step 2: It has 3 lines of symmetry through opposite vertices.
Step 3: It has 3 lines of symmetry through midpoints of opposite sides.
Step 4: Total lines of symmetry = 6
Exploring Some Geometric Themes — Practice MCQs with Answers
Attempt these 20 multiple-choice questions after working through the solutions above. Each carries the correct option and the reasoning behind it, so a wrong answer tells you which idea to revisit.
MCQ 1. What is the sum of the interior angles of a quadrilateral?
- A. 180 degrees
- B. 270 degrees
- C. 360 degrees
- D. 540 degrees
Answer: C. 360 degrees — The sum of the interior angles of any polygon with 'n' sides is given by the formula (n-2) × 180°. For a quadrilateral, n=4, so (4-2) × 180° = 2 × 180° = 360°.
MCQ 2. What is the sum of the exterior angles of any convex polygon?
- A. 90 degrees
- B. 180 degrees
- C. 360 degrees
- D. Varies with the number of sides
Answer: C. 360 degrees — The sum of the exterior angles of any convex polygon, regardless of the number of its sides, is always 360 degrees.
MCQ 3. How many diagonals does a regular hexagon have?
- A. 3
- B. 6
- C. 9
- D. 12
Answer: C. 9 — The number of diagonals in a polygon with 'n' sides is given by the formula n(n-3)/2. For a hexagon, n=6, so 6(6-3)/2 = 6 × 3 / 2 = 9 diagonals.
MCQ 4. If the interior angle of a regular polygon is 108 degrees, how many sides does it have?
- A. 4
- B. 5
- C. 6
- D. 8
Answer: B. 5 — The exterior angle of the polygon is 180° - 108° = 72°. Since the sum of exterior angles is 360°, the number of sides 'n' = 360°/72° = 5. So, it is a pentagon.
MCQ 5. Find the measure of each exterior angle of a regular octagon.
- A. 30 degrees
- B. 45 degrees
- C. 60 degrees
- D. 90 degrees
Answer: B. 45 degrees — A regular octagon has 8 equal sides and 8 equal exterior angles. The sum of exterior angles is 360 degrees, so each exterior angle is 360°/8 = 45°.
MCQ 6. In a parallelogram ABCD, if ∠A = 70 degrees, what is the measure of ∠B?
- A. 70 degrees
- B. 110 degrees
- C. 90 degrees
- D. 180 degrees
Answer: B. 110 degrees — Adjacent angles in a parallelogram are supplementary, meaning their sum is 180 degrees. So, ∠B = 180° - ∠A = 180° - 70° = 110°.
MCQ 7. Which of the following statements is always true for a parallelogram?
- A. Diagonals are equal
- B. All angles are 90 degrees
- C. Opposite sides are equal and parallel
- D. Adjacent sides are equal
Answer: C. Opposite sides are equal and parallel — By definition, a parallelogram has opposite sides that are equal in length and parallel. The other options are true only for special types of parallelograms (rectangle, square, rhombus).
MCQ 8. If the diagonals of a rectangle are (2x+1) cm and (3x-5) cm, find the value of x.
- A. 4
- B. 5
- C. 6
- D. 7
Answer: C. 6 — The diagonals of a rectangle are equal in length. Therefore, 2x + 1 = 3x - 5. Solving for x, we get 3x - 2x = 1 + 5, which means x = 6.
MCQ 9. The diagonals of a rhombus are ________ to each other.
- A. equal
- B. parallel
- C. perpendicular
- D. bisect at 30 degrees
Answer: C. perpendicular — One of the key properties of a rhombus is that its diagonals bisect each other at right angles (are perpendicular to each other).
MCQ 10. A square is a special type of ________.
- A. Trapezium only
- B. Kite only
- C. Rhombus and Rectangle
- D. Parallelogram only
Answer: C. Rhombus and Rectangle — A square possesses all the properties of a rhombus (all sides equal, diagonals perpendicular bisectors) and a rectangle (all angles 90 degrees, diagonals equal). Therefore, it is both a rhombus and a rectangle.
MCQ 11. A quadrilateral having exactly one pair of parallel sides is called a ________.
- A. Parallelogram
- B. Rhombus
- C. Trapezium (or Trapezoid)
- D. Kite
Answer: C. Trapezium (or Trapezoid) — A trapezium (or trapezoid) is defined as a quadrilateral with at least one pair of parallel sides. When it has exactly one pair of parallel sides, it is a non-isosceles trapezium.
MCQ 12. In a parallelogram PQRS, if PQ = 8 cm, then SR = ?
- A. 4 cm
- B. 6 cm
- C. 8 cm
- D. 16 cm
Answer: C. 8 cm — In a parallelogram, opposite sides are equal in length. Since PQ and SR are opposite sides, SR = PQ = 8 cm.
MCQ 13. Two adjacent angles of a parallelogram are in the ratio 2:3. Find the measure of the smaller angle.
- A. 36 degrees
- B. 72 degrees
- C. 108 degrees
- D. 180 degrees
Answer: B. 72 degrees — Adjacent angles of a parallelogram are supplementary. Let the angles be 2x and 3x. So, 2x + 3x = 180°, which gives 5x = 180°, and x = 36°. The angles are 2 × 36° = 72° and 3 × 36° = 108°. The smaller angle is 72°.
MCQ 14. The perimeter of a parallelogram is 30 cm. If one side is 4 cm longer than the other, find the lengths of its sides.
- A. 5 cm, 9 cm
- B. 5.5 cm, 9.5 cm
- C. 6 cm, 10 cm
- D. 7 cm, 11 cm
Answer: B. 5.5 cm, 9.5 cm — Let the shorter side be x cm. The longer side is (x+4) cm. The perimeter P = 2(length + breadth). So, 30 = 2(x + x+4) => 30 = 2(2x+4) => 15 = 2x+4 => 2x = 11 => x = 5.5 cm. The sides are 5.5 cm and 5.5+4 = 9.5 cm.
MCQ 15. If one angle of a rhombus is 60 degrees, what are the other angles?
- A. 60°, 60°, 60°
- B. 60°, 120°, 60°
- C. 60°, 120°, 120°
- D. 120°, 120°, 60°
Answer: C. 60°, 120°, 120° — In a rhombus, opposite angles are equal, and adjacent angles are supplementary. If one angle is 60°, its opposite angle is also 60°. The adjacent angles are 180° - 60° = 120°. So, the angles are 60°, 120°, 60°, 120°.
MCQ 16. The diagonals of a rhombus are 16 cm and 12 cm. Find the perimeter of the rhombus.
- A. 10 cm
- B. 20 cm
- C. 30 cm
- D. 40 cm
Answer: D. 40 cm — The diagonals of a rhombus bisect each other at right angles. Half the lengths of the diagonals are 16/2 = 8 cm and 12/2 = 6 cm. Using the Pythagorean theorem, the side of the rhombus (hypotenuse) = √(8² + 6²) = √(64 + 36) = √100 = 10 cm. The perimeter is 4 × side = 4 × 10 = 40 cm.
MCQ 17. Which of the following is NOT a property of a parallelogram?
- A. Opposite sides are parallel.
- B. Opposite angles are equal.
- C. Diagonals bisect each other.
- D. Diagonals are perpendicular.
Answer: D. Diagonals are perpendicular. — Diagonals being perpendicular is a property specific to a rhombus (and square), not all parallelograms. All other options are general properties of any parallelogram.
MCQ 18. The angles of a quadrilateral are x, (x+10), (x+20), and (x+30) degrees. Find the value of x.
- A. 60 degrees
- B. 75 degrees
- C. 90 degrees
- D. 100 degrees
Answer: B. 75 degrees — The sum of the angles in a quadrilateral is 360°. So, x + (x+10) + (x+20) + (x+30) = 360°. This simplifies to 4x + 60 = 360°. Subtracting 60 from both sides gives 4x = 300°, so x = 300/4 = 75°.
MCQ 19. A quadrilateral whose diagonals are equal and bisect each other perpendicularly is a ________.
- A. Rhombus
- B. Rectangle
- C. Square
- D. Kite
Answer: C. Square — Equal diagonals are a property of rectangles. Diagonals bisecting each other perpendicularly is a property of a rhombus. A figure with both properties is a square.
MCQ 20. What is the maximum number of obtuse angles a quadrilateral can have?
- A. 1
- B. 2
- C. 3
- D. 4
Answer: C. 3 — A quadrilateral can have at most three obtuse angles. If it had four obtuse angles (each > 90°), their sum would be > 360°, which is impossible for a quadrilateral.
How to Use These Solutions
Attempt each question yourself first and only then compare with the worked answer. Marks in the CBSE board exam are awarded for the METHOD as much as the final result, so reproduce the steps rather than memorising the last line. Where a solution states a law, a formula or a definition, learn that wording — examiners look for it.
Related practice for this chapter: chapter MCQs, previous-year questions, revision notes and sample papers.