Class 11 Notes - Constructions

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Constructions – Class 11

In this chapter, students learn the principles and methods of geometric constructions using only a ruler and compass. Constructions are a fundamental part of geometry, helping to visualize and solve problems accurately.

Key Concepts

  • Basic construction tools: ruler (straightedge), compass, pencil, protractor (for verification).
  • Constructing a line segment of a given length.
  • Constructing the perpendicular bisector of a line segment.
  • Constructing the bisector of a given angle.
  • Constructing angles of special measures (30°, 45°, 60°, 90°, 120°, etc.).
  • Copying a given angle.
  • Constructing triangles given different sets of data (SSS, SAS, ASA, RHS).
  • Constructing tangents to a circle from a point outside it.
  • Division of a line segment in a given ratio.
  • Constructing circles passing through given points.

Step-by-Step Examples

  1. To construct a perpendicular bisector of a line segment AB:
    1. Draw 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 the same radius, draw arcs to intersect the previous arcs at points P and Q.
    4. Draw line PQ. PQ is the perpendicular bisector of AB.
  2. To construct an angle of 60° at a point A on a line:
    1. Draw a line and mark a point A on it.
    2. With A as center, draw an arc cutting the line at B.
    3. With B as center and the same radius, draw an arc cutting the previous arc at C.
    4. Join AC. ∠BAC = 60°.
  3. To construct a triangle given three sides (SSS):
    1. Draw the base of the triangle equal to one side.
    2. With each endpoint as center, draw arcs equal to the other two sides.
    3. The intersection of arcs gives the third vertex. Join the vertices to complete the triangle.

Practice Problems

  • Construct a triangle with sides 5 cm, 6 cm, and 7 cm.
  • Construct an angle of 90° and bisect it.
  • Draw a circle of radius 4 cm and construct a tangent from a point 7 cm away from the center.
  • Divide a line segment of 8 cm in the ratio 3:2.
  • Construct a triangle given two angles (45°, 60°) and the included side of 5 cm.

Tips for Accurate Constructions

  • Always use a sharp pencil for precision.
  • Do not erase construction arcs and lines; they help in verification.
  • Label all important points clearly.
  • Practice regularly to improve accuracy and speed.

Summary

  • Geometric constructions use only a ruler and compass.
  • They help visualize and solve geometric problems accurately.
  • Common constructions include bisectors, angles, triangles, and tangents.
  • Accuracy and neatness are essential for good constructions.