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Mathematics Class 11 - Constructions Notes

Comprehensive study notes for Class 11 - Constructions olympiad preparation

Constructions

Welcome to the chapter on Constructions for Class 11. In this chapter, you will learn how to construct various geometric figures using only a ruler and compass. By the end of this chapter, you will be able to construct angles, triangles, perpendicular bisectors, tangents, and more with precision and accuracy.

Key Concepts

  • Geometric Construction: Drawing shapes, angles, and lines accurately using only a compass and straightedge (ruler).
  • Bisector: A line that divides a segment or angle into two equal parts.
  • Tangent: A line that touches a circle at exactly one point.

Basic Constructions

  • Constructing the bisector of a given angle.
  • Constructing the perpendicular bisector of a line segment.
  • Constructing an angle of 60°, 90°, 120°, etc.
  • Copying a given angle.

Constructing Triangles

  • Constructing a triangle given its three sides (SSS).
  • Constructing a triangle given two sides and the included angle (SAS).
  • Constructing a triangle given two angles and the included side (ASA).

Constructions Involving Circles

  • Constructing the tangent to a circle from a point outside the circle.
  • Constructing the circumcircle of a triangle.
  • Constructing the incircle of a triangle.

Practice Questions

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

Challenge Yourself

  • Construct a triangle given two sides and the angle opposite to one of them (SSA case, if possible).
  • Construct the incircle and circumcircle of a given triangle and mark their centers.

Did You Know?

  • Geometric constructions have been used since ancient Greek times for solving mathematical problems.
  • Some constructions, like trisecting an angle or doubling a cube, are impossible using only a compass and straightedge!

Glossary

  • Compass: A tool used to draw circles and arcs.
  • Straightedge: A ruler without markings, used to draw straight lines.
  • Bisector: A line dividing something into two equal parts.
  • Tangent: A line that touches a circle at only one point.

Answers to Practice Questions

  1. Draw the segment, place the compass at each end, draw arcs above and below, and join the intersection points to get the perpendicular bisector.
  2. Construct a 60° angle, then bisect the 120° angle to get 90°.
  3. Use the SSS construction method: draw the base, use compass to mark the other two sides, and complete the triangle.
  4. Construct a 60° angle, then use compass to bisect it for 30°.
  5. Draw the circle, mark the external point, join it to the center, construct the perpendicular bisector of this line, and use it to draw the tangent.

Practice constructions to improve your geometry skills and accuracy!

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