Class 6 Notes

Class 6 Mathematics Notes Chapter 14 (Practical Geometry) – Mathematics Book

Philoid

Philoid

— 7 min read

Mathematics
Detailed Notes with MCQs of Chapter 14: Practical Geometry. This chapter is fundamental, not just for your Class 6 understanding, but also because the principles of geometrical construction often appear in various government exams, testing your spatial reasoning and understanding of geometric properties. Pay close attention to the tools and the steps involved.

Chapter 14: Practical Geometry - Detailed Notes for Exam Preparation

1. Introduction:
Practical Geometry deals with the construction of geometric shapes using specific tools. It emphasizes accuracy and understanding the properties of shapes through the process of drawing them.

2. Tools of Construction:
Understanding the tools is the first step. You might get questions identifying the tools or their specific uses.

  • (a) The Ruler (or Straightedge):
    • Use: To draw line segments and measure their lengths.
    • Markings: Usually has centimeters (cm) and millimeters (mm) on one edge, sometimes inches on the other. A straightedge just has a straight edge with no markings.
  • (b) The Compasses:
    • Use: To draw arcs and circles. Crucially, it's used to mark off equal lengths without measuring (like copying a line segment or constructing perpendicular bisectors/angle bisectors).
    • Parts: Has two arms hinged together. One arm has a pointer, and the other has a slot to insert a pencil.
  • (c) The Divider:
    • Use: To compare lengths of line segments.
    • Parts: Looks similar to compasses but has pointers on both arms.
  • (d) Set Squares:
    • Use: To draw perpendicular and parallel lines.
    • Types: Usually come in a pair:
      • 45°-45°-90° triangle
      • 30°-60°-90° triangle
  • (e) The Protractor:
    • Use: To draw and measure angles.
    • Markings: Semi-circular shape marked with degrees from 0° to 180° (usually in both clockwise and anti-clockwise directions).

3. Basic Constructions:

  • (a) Constructing a Circle:
    • Concept: A circle is a collection of points equidistant from a fixed central point.
    • Tool: Compasses.
    • Steps:
      1. Mark a point 'O' (the center).
      2. Set the compasses to the required radius (distance between pointer and pencil tip).
      3. Place the pointer on 'O'.
      4. Rotate the pencil arm completely to draw the circle.
    • Key Terms:
      • Radius: Distance from the center to any point on the circle (e.g., OA).
      • Diameter: A line segment passing through the center with endpoints on the circle (twice the radius, e.g., BOC).
      • Chord: A line segment whose endpoints lie on the circle (e.g., PQ). The diameter is the longest chord.
      • Arc: A part of the circle's circumference.
  • (b) Constructing a Line Segment:
    • Concept: A part of a line with two fixed endpoints.
    • Tools: Ruler and Pencil.
    • Steps (for a specific length, say 5 cm):
      1. Place the ruler on the paper.
      2. Mark a point 'A' at the '0' mark of the ruler.
      3. Mark another point 'B' at the '5 cm' mark.
      4. Join A and B along the ruler's edge. AB is the required line segment.
  • (c) Constructing a Copy of a Given Line Segment:
    • Concept: Creating a line segment of the same length as a given one without measuring it with the ruler scale.
    • Tools: Ruler (as straightedge), Compasses.
    • Steps (Given segment AB, copy it):
      1. Draw a line 'l' and mark a point 'P' on it.
      2. Place the compass pointer on A and the pencil tip on B (measuring AB with compasses).
      3. Without changing the compass width, place the pointer on P.
      4. Draw an arc cutting line 'l' at point 'Q'.
      5. PQ is the copy of AB.
  • (d) Constructing a Perpendicular to a Line through a Point on the Line:
    • Concept: Drawing a line that makes a 90° angle with the given line at a specific point on it.
    • Tools: Ruler, Compasses.
    • Steps (Line 'l', point P on 'l'):
      1. With P as center and any convenient radius, draw an arc intersecting 'l' at two points, A and B.
      2. With A as center and a radius greater than AP, draw an arc above (or below) 'l'.
      3. With B as center and the same radius as in step 2, draw another arc intersecting the previous arc at point Q.
      4. Join P and Q. The line PQ is perpendicular to 'l' at P. (PQ ⊥ l)
  • (e) Constructing a Perpendicular to a Line through a Point not on the Line:
    • Concept: Drawing a line from an external point that makes a 90° angle with the given line.
    • Tools: Ruler, Compasses.
    • Steps (Line 'l', point P not on 'l'):
      1. With P as center and any convenient radius, draw an arc intersecting 'l' at two points, A and B.
      2. With A as center and a radius greater than half the length of AB, draw an arc below (or above) 'l'.
      3. With B as center and the same radius as in step 2, draw another arc intersecting the previous arc at point Q.
      4. Join P and Q. The line PQ is perpendicular to 'l' from P. (PQ ⊥ l)
  • (f) Constructing the Perpendicular Bisector of a Line Segment:
    • Concept: A line that is perpendicular to the line segment and passes through its midpoint (divides it into two equal parts).
    • Tools: Ruler, Compasses.
    • Steps (Given segment AB):
      1. With A as center and a radius greater than half the length of AB, draw arcs on both sides of AB.
      2. With B as center and the same radius, draw arcs intersecting the previous arcs at points P and Q.
      3. Join P and Q. The line PQ is the perpendicular bisector of AB. It intersects AB at its midpoint M (AM = MB) and PQ ⊥ AB.
    • Property: Any point on the perpendicular bisector of a line segment is equidistant from the endpoints of the segment (e.g., PA = PB, QA = QB).
  • (g) Constructing an Angle of a Given Measure:
    • Concept: Creating an angle with a specific degree value.
    • Tools: Ruler, Protractor.
    • Steps (e.g., Construct 70° angle with vertex O):
      1. Draw a ray OA (one arm of the angle).
      2. Place the center of the protractor at O and the baseline along OA.
      3. Find the 70° mark on the scale (starting from 0° on OA's side). Mark a point B at 70°.
      4. Remove the protractor and join O to B. ∠AOB is the required 70° angle.
  • (h) Constructing a Copy of an Angle of Unknown Measure:
    • Concept: Replicating a given angle without using a protractor.
    • Tools: Ruler (as straightedge), Compasses.
    • Steps (Given ∠ABC, copy it with vertex P on ray PQ):
      1. Place compass pointer at vertex B of the given angle and draw an arc cutting arms BA and BC at points D and E respectively.
      2. With the same compass setting, place the pointer at P and draw a similar arc cutting ray PQ at point R.
      3. Set the compass width equal to the distance DE (measure the opening of the angle).
      4. With R as center and the compass width DE, draw an arc intersecting the arc drawn in step 2 at point S.
      5. Join P to S. ∠SPQ is the copy of ∠ABC.
  • (i) Constructing the Bisector of an Angle:
    • Concept: A ray that divides an angle into two equal angles.
    • Tools: Ruler, Compasses.
    • Steps (Given ∠ABC):
      1. With B as center and any convenient radius, draw an arc intersecting arms BA and BC at points D and E respectively.
      2. With D as center and a radius greater than half the length DE, draw an arc in the interior of the angle.
      3. With E as center and the same radius as in step 2, draw another arc intersecting the first arc at point F.
      4. Join B to F. Ray BF is the angle bisector of ∠ABC (i.e., ∠ABF = ∠FBC = ½ ∠ABC).
  • (j) Constructing Angles of Special Measures (Using Ruler and Compasses Only):
    • 60°:
      1. Draw a ray OA.
      2. With O as center and any radius, draw an arc cutting OA at P.
      3. With P as center and the same radius, draw an arc cutting the first arc at Q.
      4. Join OQ. ∠AOQ = 60°. (Based on equilateral triangle properties).
    • 30°: Construct a 60° angle and then bisect it.
    • 120°:
      1. Construct a 60° angle (∠AOQ = 60° as above).
      2. With Q as center and the same radius (OP), draw another arc cutting the initial large arc at R.
      3. Join OR. ∠AOR = 120°. (Effectively 60° + 60°).
    • 90°:
      Method 1: Construct a perpendicular on a line at a point (as described earlier).
      Method 2: Construct 60° and 120° angles sharing the same vertex and initial arm. Bisect the angle between the 60° and 120° marks (i.e., bisect the second 60° angle).
      Method 3: Construct a 60° angle, then construct another 60° adjacent to it (making 120°). Bisect the second 60° angle. The line will be at 60° + 30° = 90°.
    • 45°: Construct a 90° angle and then bisect it.

4. Key Takeaways for Exams:

  • Know the precise use of each tool.
  • Memorize the steps for each basic construction. Questions might ask about a specific step or the sequence.
  • Understand the properties related to constructions (e.g., perpendicular bisector properties, angle bisector properties).
  • Practice identifying angles that can be constructed using only ruler and compasses (multiples/fractions of 60° and 90° like 15°, 22.5°, 30°, 45°, 60°, 75°, 90°, 105°, 120°, 135°, 150°, etc.).

Multiple Choice Questions (MCQs):

  1. Which instrument is used to draw circles and arcs?
    (a) Ruler
    (b) Protractor
    (c) Compasses
    (d) Set Square

  2. A line that divides a line segment into two equal parts and is perpendicular to it is called:
    (a) Angle bisector
    (b) Perpendicular bisector
    (c) Median
    (d) Altitude

  3. To construct a copy of a given line segment AB using compasses, after drawing a line 'l' and marking point P, what is the first step involving the compasses?
    (a) Draw an arc from P with any radius.
    (b) Measure the length of AB using the compasses.
    (c) Measure the length of AB using a ruler.
    (d) Draw a perpendicular at P.

  4. Which of the following angles CANNOT be constructed using only a ruler and compasses?
    (a) 30°
    (b) 45°
    (c) 70°
    (d) 90°

  5. While constructing the perpendicular bisector of a line segment XY, the radius of the arc drawn from X should be:
    (a) Equal to XY
    (b) Less than half of XY
    (c) Equal to half of XY
    (d) Greater than half of XY

  6. To bisect an angle, after drawing an initial arc cutting the arms, what is the next step?
    (a) Draw another arc from the vertex with a different radius.
    (b) Draw intersecting arcs from the points where the first arc cut the arms.
    (c) Measure the angle using a protractor.
    (d) Draw a parallel line to one of the arms.

  7. The instrument used to measure or draw angles of specific degrees is:
    (a) Compasses
    (b) Set Square
    (c) Protractor
    (d) Divider

  8. Constructing a 60° angle with ruler and compasses relies on the properties of which shape?
    (a) Square
    (b) Rectangle
    (c) Isosceles Triangle
    (d) Equilateral Triangle

  9. If you construct a 90° angle and then bisect it, what angle measure will you get?
    (a) 30°
    (b) 45°
    (c) 60°
    (d) 180°

  10. What is the primary function of a divider in practical geometry?
    (a) To draw arcs
    (b) To measure angles
    (c) To compare lengths
    (d) To draw perpendicular lines

Answer Key for MCQs:

  1. (c)
  2. (b)
  3. (b)
  4. (c)
  5. (d)
  6. (b)
  7. (c)
  8. (d)
  9. (b)
  10. (c)

Study these notes carefully, focusing on the steps and the logic behind them. Good luck with your preparation!

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