Class 9 Notes - Constructions

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1. Introduction to Constructions

Geometric constructions are precise drawings of shapes, angles, and lines using only a ruler and a compass. In Class 9, students learn to construct various geometric figures following mathematical rules and logic.

2. Basic Tools for Construction

  • Ruler (Scale): For drawing straight lines and measuring lengths.
  • Compass: For drawing arcs and circles.
  • Protractor: For measuring and drawing angles (sometimes allowed for checking).
  • Pencil and Eraser: For neat and accurate drawings.

3. Fundamental Constructions

  • Bisecting a given angle
  • Bisecting a given line segment
  • Constructing a perpendicular bisector of a line segment
  • Constructing an angle of 60°, 90°, 30°, 45°, etc.
  • Constructing a triangle given different sets of data (SSS, SAS, ASA, RHS)
  • Constructing tangents to a circle from a point outside it

4. Step-by-Step Examples

4.1 Bisecting a Line Segment

  1. Draw a line segment AB.
  2. With A as center and radius more than half of AB, draw arcs above and below AB.
  3. With B as center and same radius, draw arcs to cut the previous arcs at P and Q.
  4. Draw line PQ. It bisects AB at point M.

4.2 Constructing a 60° Angle

  1. Draw a line and mark a point O on it.
  2. With O as center, draw an arc cutting the line at A.
  3. With A as center and same radius, draw an arc to cut the previous arc at B.
  4. Join OB. ∠AOB = 60°.

4.3 Constructing a Triangle (Given Sides)

  1. Draw the base of the triangle as per the given length.
  2. With each endpoint as center, draw arcs with radii equal to the other two sides.
  3. The intersection of arcs gives the third vertex. Join to form the triangle.

4.4 Constructing Tangents to a Circle

  1. Draw a circle with center O and a point P outside the circle.
  2. Join OP. Find the midpoint M of OP.
  3. With M as center and radius MO, draw a circle. It cuts the given circle at points Q and R.
  4. Join PQ and PR. These are the tangents from P to the circle.

5. Tips for Accurate Construction

  • Always use sharp pencils for clear lines and arcs.
  • Do not erase construction lines unless asked.
  • Label all important points clearly.
  • Use the compass carefully to avoid slipping.

6. Common Mistakes

  • Not keeping the compass radius fixed while drawing arcs.
  • Not measuring accurately with the ruler.
  • Forgetting to label points or lines.
  • Erasing necessary construction lines.

7. Practice Exercises

  1. Construct a triangle with sides 5 cm, 6 cm, and 7 cm.
  2. Construct an angle of 90° at a given point on a line.
  3. Bisect a given angle of 60°.
  4. Draw a circle of radius 4 cm and construct a tangent from a point 7 cm away from the center.
  5. Construct a perpendicular bisector of a line segment 8 cm long.

8. Summary

  • Constructions require accuracy and understanding of geometric principles.
  • Practice helps in mastering the use of compass and ruler.
  • Always follow the steps methodically for correct results.