Class 8 Mathematics Notes Chapter 4 (Practical Geometry) – Mathematics Book

Alright class, let's focus on Chapter 4: Practical Geometry. This chapter is fundamental, not just for your Class 8 exams, but it also builds a base for geometry concepts often tested in various government exams. The core idea is understanding what minimum information is required to draw a unique geometrical figure, specifically quadrilaterals.

Chapter 4: Practical Geometry - Detailed Notes

1. Introduction: Why Practical Geometry?

• This chapter teaches the step-by-step process of drawing geometric shapes accurately using tools like a ruler, compass, and protractor.
• The key focus is on constructing quadrilaterals when specific measurements (sides, diagonals, angles) are provided.
• Uniqueness: The goal is to construct a unique quadrilateral. This means anyone following the steps with the given measurements should end up with the exact same shape and size (congruent figures).

2. Recap: Constructing Triangles

• Remember, to construct a unique triangle, we needed 3 measurements (in specific combinations):
  • SSS (Side-Side-Side)
  • SAS (Side-Angle-Side)
  • ASA (Angle-Side-Angle)
  • RHS (Right angle-Hypotenuse-Side)

3. Constructing Quadrilaterals: The Need for More Information

• A quadrilateral has 4 sides and 4 angles (8 parameters) + 2 diagonals (total 10 parameters).
• Unlike triangles, giving just 3 or even 4 measurements is often not enough to construct a unique quadrilateral. For example, if you only know the 4 side lengths, you can often flex the shape (like a rhombus changing angles).
• Generally, we need 5 independent measurements to define a unique quadrilateral.

4. Methods for Constructing a Unique Quadrilateral:

The chapter systematically explores different combinations of 5 measurements:

(i) When Four Sides and One Diagonal are Given (SSSS D)

• Concept: Knowing one diagonal splits the quadrilateral into two triangles. We can construct these triangles using the SSS criterion.
• Strategy:
  1. Use the diagonal and two sides forming one triangle. Construct this triangle using SSS (draw the diagonal, then use arcs from its endpoints for the other two sides).
  2. Use the diagonal and the remaining two sides forming the second triangle. Locate the fourth vertex using arcs from the endpoints of the diagonal.
  3. Join the vertices to complete the quadrilateral.
• Example: Construct quadrilateral ABCD where AB, BC, CD, DA, and diagonal AC are known.
  • Construct ΔABC using SSS (AB, BC, AC).
  • Construct ΔADC using SSS (AD, DC, AC). Point D is found by arcs from A (radius AD) and C (radius CD).
  • Join AD and CD.

(ii) When Two Diagonals and Three Sides are Given (SSD DD)

• Concept: This usually involves constructing one triangle using SSS first, and then using the lengths of the diagonals and the remaining side to locate the fourth vertex.
• Strategy:
  1. Identify three sides that form a triangle (e.g., if sides AB, BC, CD and diagonals AC, BD are given, you can start with ΔABC using AB, BC, and AC if AC is known, or ΔBCD using BC, CD, and BD if BD is known). Let's assume AC is known, so we use AB, BC, CD, AC, BD.
  2. Construct the triangle using the three known sides (e.g., ΔABC using AB, BC, and AC via SSS).
  3. Now, locate the fourth vertex (D). We know the length CD and BD.
  4. From C, draw an arc with radius CD.
  5. From B, draw an arc with radius BD.
  6. The intersection of these arcs gives point D.
  7. Join AD and CD.
• Note: Careful selection of the initial triangle is important based on the sides provided.

(iii) When Two Adjacent Sides and Three Angles are Given (ASASA type pattern)

• Concept: We use the angles to define the direction of the sides. The Angle Sum Property of a quadrilateral (sum of angles is 360°) might be needed if the included angle isn't directly given.
• Strategy:
  1. Draw one of the given adjacent sides (say AB).
  2. Construct the angle at one endpoint (e.g., ∠B). Draw a ray.
  3. Measure the second adjacent side along this ray (mark point C if BC is the second side).
  4. Construct the angle at the other endpoint of the first side (e.g., ∠A). Draw a ray.
  5. Construct the angle at the newly found vertex (e.g., ∠C). Draw a ray.
  6. The intersection of the rays from A and C gives the fourth vertex (D).
• Example: Construct ABCD where AB, BC, ∠A, ∠B, ∠C are known.
  • Draw AB.
  • Construct ∠B at B. Mark C on the ray such that BC is the given length.
  • Construct ∠A at A. Draw a ray AX.
  • Construct ∠C at C. Draw a ray CY.
  • The intersection of AX and CY is D.

(iv) When Three Sides and Two Included Angles are Given (SASAS type pattern)

• Concept: Similar to SAS for triangles, the angles are included between the given sides.
• Strategy:
  1. Draw one of the sides which has angles given at both its ends (e.g., side BC if ∠B and ∠C are given).
  2. Construct the angle at one endpoint (e.g., ∠B).
  3. Measure the side length connected to this angle (e.g., AB) along the ray and mark the vertex (A).
  4. Construct the angle at the other endpoint of the initial side (e.g., ∠C).
  5. Measure the side length connected to this angle (e.g., CD) along the ray and mark the vertex (D).
  6. Join the two newly found vertices (A and D).
• Example: Construct ABCD where AB, BC, CD, ∠B, ∠C are known.
  • Draw BC.
  • Construct ∠B at B. Mark A on the ray such that BA = given length.
  • Construct ∠C at C. Mark D on the ray such that CD = given length.
  • Join AD.

5. Special Cases: Construction of Specific Quadrilaterals

• Sometimes, you need to construct special quadrilaterals like squares, rectangles, rhombuses, or parallelograms. Their properties reduce the number of explicit measurements needed.
• Square:
  • Property: All sides equal, all angles 90°.
  • Minimum needed: Just 1 side length is enough. (Construct a 90° angle, mark sides, complete). Or 1 diagonal length (Diagonals are equal and bisect at 90°).
• Rectangle:
  • Property: Opposite sides equal, all angles 90°.
  • Minimum needed: 2 adjacent side lengths. (Construct 90° angles).
• Rhombus:
  • Property: All sides equal, opposite angles equal, diagonals bisect each other at 90°.
  • Minimum needed: 2 diagonals OR 1 side and 1 angle.
• Parallelogram:
  • Property: Opposite sides equal and parallel, opposite angles equal, diagonals bisect each other.
  • Minimum needed: 2 adjacent sides and the included angle OR 2 adjacent sides and 1 diagonal OR 2 diagonals and the angle between them.

Key Takeaways for Exams:

• Know the minimum 5 measurements generally required for a unique quadrilateral.
• Understand the 4 standard construction methods (SSSS D, SSD DD, ASASA type, SASAS type).
• Be able to apply the properties of special quadrilaterals to determine the minimum required measurements for their construction.
• Practice the steps using a compass and ruler. Even for MCQs, visualizing the construction process helps.
• Remember the Angle Sum Property of Quadrilaterals (360°).

Multiple Choice Questions (MCQs)

Here are 10 MCQs based on Chapter 4, suitable for government exam preparation:

1. What is the minimum number of independent measurements required to construct a unique quadrilateral?
   (a) 3
   (b) 4
   (c) 5
   (d) 6

2. To construct a unique parallelogram, the minimum number of measurements needed is:
   (a) 2 adjacent sides
   (b) 2 adjacent sides and 1 diagonal
   (c) 4 sides
   (d) 1 diagonal and 1 side

3. Which set of measurements is sufficient to construct a unique quadrilateral ABCD?
   (a) AB, BC, CD, DA
   (b) AB, BC, CD, ∠B
   (c) AB, BC, CD, DA, AC
   (d) AC, BD, ∠A, ∠B

4. To construct a unique square, what is the minimum measurement(s) required?
   (a) Two adjacent sides
   (b) One side length
   (c) Two diagonals
   (d) One side and one angle

5. If you are given the lengths of two diagonals and three sides of a quadrilateral, which construction method type is applicable?
   (a) SSSS D
   (b) SSD DD
   (c) ASASA type
   (d) SASAS type

6. To construct a rhombus uniquely, which of the following is sufficient?
   (a) Lengths of its two diagonals
   (b) Length of one side only
   (c) Lengths of two adjacent sides
   (d) Length of one side and one diagonal

7. Which of the following measurements is NOT sufficient to construct a unique quadrilateral?
   (a) Four sides and one diagonal
   (b) Three sides and two included angles
   (c) Two adjacent sides and three angles
   (d) Four sides and one angle

8. You are asked to construct a quadrilateral PLAN where PL=4cm, LA=6.5cm, ∠P=90°, ∠A=110°, ∠N=85°. What is the measure of ∠L?
   (a) 75°
   (b) 85°
   (c) 90°
   (d) Insufficient information

9. To construct a rectangle uniquely, the minimum measurements needed are:
   (a) One side length
   (b) One diagonal length
   (c) Lengths of two adjacent sides
   (d) Lengths of two opposite sides

10. In which construction type do you primarily rely on constructing triangles using the SSS criterion first?
    (a) When 2 adjacent sides and 3 angles are given
    (b) When 3 sides and 2 included angles are given
    (c) When 4 sides and 1 diagonal are given
    (d) When constructing a square using one side

Answer Key for MCQs:

1. (c) 5
2. (b) 2 adjacent sides and 1 diagonal (or 2 adjacent sides and included angle)
3. (c) AB, BC, CD, DA, AC (This is the SSSS D case)
4. (b) One side length (as all angles are 90°)
5. (b) SSD DD
6. (a) Lengths of its two diagonals (They bisect at 90°)
7. (d) Four sides and one angle (This can often lead to more than one possible shape unless it's a special quadrilateral or the angle is included). Four sides alone are definitely not enough.
8. (a) 75° (Sum of angles in a quadrilateral is 360°. ∠L = 360° - (90° + 110° + 85°) = 360° - 285° = 75°)
9. (c) Lengths of two adjacent sides (as all angles are 90°)
10. (c) When 4 sides and 1 diagonal are given (The diagonal splits it into two triangles defined by SSS)

Study these notes well and practice the constructions. Good luck!