Class 7 Mathematics Notes Chapter 10 (Practical Geometry) – Mathematics Book

Alright class, let's get straight into Chapter 10: Practical Geometry. This chapter is crucial not just for your school exams but also forms a foundation for geometry questions in various government exams. It's all about constructing geometric shapes accurately using basic instruments.

Chapter 10: Practical Geometry - Detailed Notes

1. Introduction

• Practical Geometry deals with the methods of drawing geometric figures accurately using specific tools.
• Primary Tools Used:
  • Ruler (Straightedge): For drawing line segments and measuring their lengths.
  • Compasses: For drawing arcs, circles, and marking off equal lengths without measuring.
  • Protractor: For measuring and drawing angles (though many constructions aim to use only ruler and compasses).
  • Set-squares: For drawing parallel and perpendicular lines (often used as aids).
• Accuracy is key in these constructions. Use a sharp pencil and precise measurements.

2. Construction of a Line Parallel to a Given Line Through a Point Not On It

• Objective: Given a line 'l' and a point 'P' outside it, construct a line 'm' through 'P' such that m || l.
• Underlying Principle: If two lines are intersected by a transversal such that a pair of alternate interior angles (or corresponding angles) are equal, then the lines are parallel.
• Steps (Using Alternate Interior Angles Property):
  1. Take any point 'A' on the given line 'l'.
  2. Join point 'P' to point 'A'. This line segment PA is the transversal.
  3. With 'A' as the centre and any convenient radius, draw an arc cutting line 'l' at 'B' and the transversal PA at 'C'.
  4. With 'P' as the centre and the same radius as in Step 3, draw another arc cutting the transversal PA at 'D'. Let this arc be EF.
  5. Now, adjust the compasses to measure the distance BC (the opening of the first arc).
  6. With 'D' as the centre and the compass opening equal to BC (from Step 5), draw an arc to cut the arc EF at point 'Q'.
  7. Join point 'P' to point 'Q' and extend it to form the line 'm'.
  8. Line 'm' is parallel to line 'l'. (Because ∠QPA = ∠CAB, which are alternate interior angles).

3. Construction of Triangles

• A triangle can be constructed uniquely if the measures of certain elements (sides and angles) are given, corresponding to the congruence criteria.
• Prerequisite Knowledge:
  • Triangle Inequality Property: The sum of the lengths of any two sides of a triangle must be greater than the length of the third side. (e.g., a + b > c, b + c > a, a + c > b). This must be checked before attempting SSS construction.
  • Angle Sum Property: The sum of angles in a triangle is 180°. This is relevant for ASA construction (the two given angles must sum to less than 180°).

3.1 Construction using SSS (Side-Side-Side) Criterion

• Objective: Construct a triangle when the lengths of all three sides are known.
• Condition: The sum of any two sides must be greater than the third side.
• Steps (Example: Construct ΔABC with AB = 5 cm, BC = 6 cm, AC = 4 cm):
  1. Draw a line segment BC of length 6 cm (usually the longest side is taken as the base, but any side works).
  2. With 'B' as the centre and radius 5 cm (length of AB), draw an arc.
  3. With 'C' as the centre and radius 4 cm (length of AC), draw another arc that intersects the previous arc at point 'A'.
  4. Join AB and AC.
  5. ΔABC is the required triangle.

3.2 Construction using SAS (Side-Angle-Side) Criterion

• Objective: Construct a triangle when the lengths of two sides and the measure of the included angle are known. (The angle must be between the two given sides).
• Steps (Example: Construct ΔPQR with PQ = 4 cm, QR = 5 cm, ∠PQR = 60°):
  1. Draw a line segment QR of length 5 cm.
  2. At point 'Q', construct an angle ∠RQX = 60° using compasses or a protractor. (Compasses preferred: draw an arc from Q, cut it with the same radius for 60°).
  3. With 'Q' as the centre and radius 4 cm (length of PQ), draw an arc cutting the ray QX at point 'P'.
  4. Join PR.
  5. ΔPQR is the required triangle.

3.3 Construction using ASA (Angle-Side-Angle) Criterion

• Objective: Construct a triangle when the measures of two angles and the length of the included side are known. (The side must be between the two given angles).
• Condition: The sum of the two given angles must be less than 180°.
• Steps (Example: Construct ΔXYZ with XY = 6 cm, ∠ZXY = 30°, ∠ZYX = 100°):
  1. Draw a line segment XY of length 6 cm.
  2. At point 'X', construct an angle ∠YXA = 30°.
  3. At point 'Y', construct an angle ∠XYB = 100°.
  4. Let the rays XA and YB intersect at point 'Z'.
  5. ΔXYZ is the required triangle.
  • Note: If two angles and a non-included side are given (AAS), you can find the third angle using the angle sum property (180° - sum of given angles) and then proceed as ASA.

3.4 Construction using RHS (Right angle-Hypotenuse-Side) Criterion

• Objective: Construct a right-angled triangle when the length of the hypotenuse and one side are known.
• Steps (Example: Construct ΔLMN, right-angled at M, given LN = 5 cm and MN = 3 cm):
  1. Draw a line segment MN of length 3 cm.
  2. At point 'M', construct a perpendicular line segment MX (i.e., construct ∠XMN = 90°).
  3. With 'N' as the centre and radius 5 cm (length of the hypotenuse LN), draw an arc cutting the ray MX at point 'L'.
  4. Join LN.
  5. ΔLMN is the required right-angled triangle.

Key Takeaways for Exams:

• Understand the conditions required for each type of construction (SSS, SAS, ASA, RHS).
• Be familiar with the steps involved in each construction.
• Remember the prerequisite conditions: Triangle Inequality for SSS, Angle Sum Property for ASA.
• Know how to construct basic angles (60°, 90°, 30°, 45°, 120°) and perpendicular bisectors using ruler and compasses, as these are often intermediate steps.
• The principle behind constructing parallel lines (alternate interior or corresponding angles) is important.

Multiple Choice Questions (MCQs)

1. To construct a line parallel to a given line 'l' through a point 'P' not on 'l', we use the property that:
   a) Sum of angles on a straight line is 180°.
   b) Vertically opposite angles are equal.
   c) Alternate interior angles (or corresponding angles) are equal.
   d) The exterior angle of a triangle equals the sum of interior opposite angles.

2. Which criterion is used to construct a triangle ABC where AB = 5 cm, BC = 6 cm, and AC = 7 cm?
   a) ASA
   b) SAS
   c) SSS
   d) RHS

3. To construct a triangle PQR with PQ = 6 cm, ∠P = 60°, and PR = 5 cm, which criterion will be used?
   a) ASA
   b) SAS
   c) SSS
   d) RHS

4. Is it possible to construct a triangle with side lengths 3 cm, 4 cm, and 8 cm?
   a) Yes, using SSS criterion.
   b) Yes, using SAS criterion.
   c) No, because 3 + 4 is not greater than 8.
   d) No, because the angles are not given.

5. To construct a triangle DEF with DE = 5 cm, ∠DEF = 45°, and ∠EDF = 60°, which criterion is applicable?
   a) SSS
   b) SAS
   c) ASA (after finding ∠DFE)
   d) ASA (directly using the given information)

6. In the construction of a triangle using the ASA criterion, we are given two angles and:
   a) The side opposite to one of the angles.
   b) The side included between the two angles.
   c) Any one of the three sides.
   d) The hypotenuse.

7. Which criterion is specifically used for constructing a right-angled triangle?
   a) SSS
   b) SAS
   c) ASA
   d) RHS

8. To construct a right-angled triangle LMN, right-angled at M, with hypotenuse LN = 8 cm and side MN = 5 cm, the first step is usually:
   a) Draw the hypotenuse LN = 8 cm.
   b) Draw the side MN = 5 cm.
   c) Construct the 90° angle.
   d) Draw side LM (length unknown).

9. If you are given two angles of a triangle as 70° and 110°, can you construct the triangle?
   a) Yes, using ASA.
   b) Yes, using SSS.
   c) No, because the sum of the two angles is 180°.
   d) No, because a side length is not given.

10. When constructing a line parallel to a given line using the alternate interior angles method, after drawing the transversal and the first arc, the essential next step involving the compasses is:
    a) Drawing another arc with a different radius from the external point.
    b) Measuring the angle formed using a protractor.
    c) Keeping the same radius and drawing an arc from the external point.
    d) Drawing a perpendicular line.

Answer Key for MCQs:

1. c) Alternate interior angles (or corresponding angles) are equal.
2. c) SSS
3. b) SAS (Angle P is included between sides PQ and PR).
4. c) No, because 3 + 4 is not greater than 8 (violates Triangle Inequality Property).
5. d) ASA (The side DE is included between angles ∠EDF and ∠DEF, even though ∠EDF is written first, it's angle D, and ∠DEF is angle E, side DE is between D and E). Correction: My explanation was slightly confusing. The question gives ∠DEF (angle E) and ∠EDF (angle D). The side given is DE. This side is included between ∠D and ∠E. So ASA applies directly.
6. b) The side included between the two angles.
7. d) RHS
8. b) Draw the side MN = 5 cm (or construct the 90° angle first, then mark MN). Drawing the known side or the right angle are the typical starting points. Option (b) is a very common first step.
9. c) No, because the sum of the two angles is 180° (violates Angle Sum Property for a triangle).
10. c) Keeping the same radius and drawing an arc from the external point.

Study these notes carefully, focusing on the steps and the underlying geometric principles. Practice the constructions yourself using a ruler and compasses for better understanding. Good luck!