This is a practical lesson where you must use a pair of compasses and a straight edge (ruler without markings) to draw geometric figures accurately. In the exam, you must show all your construction arcs clearly.

1. Required Tools & Rules

• Tools: A straight edge (ruler), a pair of compasses, and a sharp pencil.

• Golden Rule: DO NOT erase your construction arcs. The examiner needs to see them to award marks. Using a pen is not allowed. All lines and arcs must be thin and precise.

2. The Four Basic Constructions (Building Blocks)

You must master these four fundamental constructions. Every complex problem is built from them.

1. Constructing the Perpendicular Bisector of a Line Segment:

   • Purpose: To find the exact midpoint of a line and draw a line at 90° through it.

   • Method:

     1. Open your compass to a radius that is more than half the length of the line segment.

     2. Place the compass point on one end of the line and draw an arc above and below the line.

     3. Without changing the compass radius, place the point on the other end and draw two more arcs to intersect the first ones.

     4. Join the two points where the arcs intersect. This new line is the perpendicular bisector.

2. Constructing the Bisector of an Angle:

   • Purpose: To divide an angle into two equal angles.

   • Method:

     1. Place the compass point on the vertex of the angle and draw an arc that cuts both arms of the angle.

     2. From the two points where the arc cuts the arms, draw two more intersecting arcs in the middle of the angle.

     3. Join the vertex to the point where these new arcs intersect.

3. Constructing a Perpendicular to a Line from a point on the line:

   • Purpose: To draw a 90° line at a specific point on an existing line.

   • Method:

     1. Place the compass point on the given point on the line.

     2. Draw two small arcs to cut the line on either side of the point, creating two new points.

     3. Use these two new points as the ends of a new line segment and construct its perpendicular bisector (as in method 1).

4. Constructing a Perpendicular to a Line from an external point:

   • Purpose: To drop a 90° line from a point down to a line.

   • Method:

     1. Place the compass point on the external point.

     2. Draw an arc that cuts the line segment in two places.

     3. Using these two new points on the line, draw two intersecting arcs on the opposite side of the line.

     4. Join the original external point to this new intersection point.

3. Key Triangle and Circle Constructions

Circumcircle of a Triangle

• What it is: The circle that passes through all three vertices of a triangle.

• How to construct it:

  1. Construct the perpendicular bisectors of any two sides of the triangle.

  2. The point where they intersect is the centre of the circle (the circumcentre).

  3. Place the compass point on the circumcentre, set the radius to any of the triangle's vertices, and draw the circle.

Incircle of a Triangle

• What it is: The circle that is drawn inside a triangle, touching all three sides.

• How to construct it:

  1. Construct the angle bisectors of any two angles of the triangle.

  2. The point where they intersect is the centre of the circle (the incentre).

  3. To find the radius, construct a perpendicular from the incentre to any of the sides.

  4. Use this perpendicular distance as the radius to draw the circle.

Tangents to a Circle

1. Through a point ON the circle:

   • Draw the radius from the centre to the point.

   • Construct a perpendicular to the radius at that point. This perpendicular line is the tangent.

2. From an EXTERNAL point:

   • Join the external point (P) to the centre of the circle (O).

   • Construct the perpendicular bisector of the line PO to find its midpoint (M).

   • With the centre at M and radius MO, draw a new circle (or arcs) that intersects your original circle at two points.

   • These two intersection points are the points of contact. Join P to these points to get the two tangents.

Exam Tips & Common Pitfalls

• Pitfall 1: Inaccuracy. A blunt pencil or wobbly compass will lead to inaccurate drawings and lost marks. Use good quality tools.

• Pitfall 2: Not showing arcs. If you measure a 90° angle with a protractor, you will get zero marks. You must show the construction arcs to prove you used the correct method.

• Exam Tip 1: Constructing special angles. You are expected to construct angles like 90°, 45°, 60°, and 30°.

  • 60°: Construct an equilateral triangle.

  • 90°: Construct a perpendicular.

  • 45°: Bisect a 90° angle.

  • 30°: Bisect a 60° angle.

• Exam Tip 2: Follow the sequence. Construction questions in the exam are often multi-part. Read the whole question first, but complete it step-by-step in the order given. The result of one part is usually needed for the next.