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Mathematics Class 9 - Constructions Notes

Comprehensive study notes for Class 9 - Constructions olympiad preparation

Constructions

In this chapter, you will learn how to construct geometric shapes and figures using a ruler and compass. You will understand the steps for drawing lines, angles, triangles, and other figures accurately. By the end of this chapter, you will be able to follow construction steps and solve geometry problems with confidence.

Key Concepts

  • Construction: Drawing shapes, angles, and figures using only a ruler and compass.
  • Perpendicular bisector: A line that divides another line into two equal parts at 90°.
  • Angle bisector: A line that divides an angle into two equal angles.

Basic Constructions

  • Constructing a perpendicular bisector of a line segment.
  • Constructing the bisector of a given angle.
  • Constructing angles of 30°, 45°, 60°, 90°, and 120° using compass and ruler.
  • Constructing a triangle when its sides or angles are given.

Steps for Construction

  • Draw a line segment of given length using a ruler.
  • Use a compass to draw arcs and mark points.
  • Join points to form angles or triangles as required.
  • Label all points and lines clearly.

Example: Constructing a Triangle

Example: Construct a triangle ABC with AB = 5 cm, BC = 6 cm, and CA = 7 cm.

  • Draw AB = 5 cm.
  • With A as center, draw an arc of radius 7 cm.
  • With B as center, draw an arc of radius 6 cm to intersect the previous arc at C.
  • Join AC and BC to complete the triangle.

Applications

  • Constructions are used in geometry, engineering, and design.
  • They help in drawing accurate diagrams for proofs and solutions.

Practice Questions

  1. Construct a perpendicular bisector of a line segment of length 8 cm.
  2. Draw an angle of 60° using a compass and ruler.
  3. Construct a triangle with sides 4 cm, 5 cm, and 6 cm.
  4. Construct the bisector of an angle of 90°.
  5. Explain the steps to construct a triangle when two sides and the included angle are given.

Challenge Yourself

  • Construct a triangle with sides 5 cm, 6 cm, and 8 cm and measure its angles.
  • Draw a perpendicular from a point to a given line using only a ruler and compass.

Did You Know?

  • Ancient mathematicians used only a compass and straightedge for constructions.
  • Constructions help in understanding geometric properties and relationships.

Glossary

  • Compass: A tool used to draw circles and arcs.
  • Ruler: A tool used to draw straight lines and measure lengths.
  • Bisector: A line that divides something into two equal parts.

Answers to Practice Questions

  1. Draw the line segment, use the compass to draw arcs from both ends, and join the intersection points to get the bisector.
  2. Draw a line, use the compass to construct an equilateral triangle, and the angle at the base will be 60°.
  3. Follow the triangle construction steps as shown in the example above.
  4. Draw the angle, use the compass to mark equal arcs, and join the vertex to the arc intersection for the bisector.
  5. Draw the first side, use the protractor to measure the angle, then draw the second side, and finally join the endpoints to complete the triangle.

Practice constructions to improve your geometry skills and accuracy!

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