Class 10 Notes - Constructions

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1. Introduction to Constructions

Geometric construction is the process of drawing shapes, angles, and lines accurately using only a compass and a straightedge (ruler without measurement markings). This chapter focuses on the fundamental constructions required in Class 10 mathematics.

2. Basic Tools for Construction

  • Compass
  • Straightedge (ruler without markings)
  • Pencil
  • Eraser

3. Fundamental Constructions

  • Bisecting a given angle
  • Bisecting a given line segment
  • Constructing a triangle given its base, a base angle, and the sum/difference of the other two sides
  • Constructing tangents to a circle from a point outside it
  • Constructing similar triangles

4. Step-by-Step Examples

4.1 Bisecting a Given Angle

  1. Draw the given angle ∠ABC.
  2. With B as center, draw an arc cutting both arms of the angle at P and Q.
  3. With P and Q as centers and the same radius, draw arcs that intersect at R.
  4. Draw a straight line from B through R. This line bisects ∠ABC.

4.2 Bisecting a Line Segment

  1. Draw the line segment AB.
  2. With A and B as centers and radius more than half of AB, draw arcs above and below the line to intersect at points P and Q.
  3. Draw line PQ. It bisects AB at point M.

4.3 Constructing a Tangent to a Circle from a Point Outside It

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

5. Important Tips

  • Always use sharp pencils for accuracy.
  • Do not use the ruler’s measurements unless specified.
  • Keep the compass steady while drawing arcs.
  • Label all important points clearly.

6. Practice Problems

  1. Construct a triangle given its base, a base angle, and the sum of the other two sides.
  2. Construct a triangle given its base, a base angle, and the difference of the other two sides.
  3. Draw a circle of radius 4 cm and construct a tangent from a point 7 cm away from the center.
  4. Bisect a given angle of 70° using only a compass and straightedge.

7. Common Mistakes

  • Using ruler measurements instead of compass and straightedge.
  • Not keeping the compass radius fixed while drawing arcs.
  • Incorrectly labeling points, leading to confusion.
  • Not drawing arcs long enough to intersect.

8. Summary

  • Geometric constructions use only a compass and straightedge.
  • Key constructions include bisecting angles and lines, constructing triangles, and drawing tangents to circles.
  • Accuracy and clear labeling are essential for correct constructions.