Constructions

Chapter in a nutshell: Using only a ruler and compass, you construct tangents to a circle, the circumscribed and inscribed circles of a triangle, and regular polygons (e.g. a hexagon). Accuracy depends on the basic constructions — perpendicular bisector, angle bisector and copying angles.

1. Key Constructions (ICSE scope)

• Tangents to a circle from an external point P: join PO, draw a circle on PO as diameter; it cuts the given circle at the points of contact; join P to them.
• Circumscribed circle (circumcircle) of a triangle: centre = circumcentre = intersection of the perpendicular bisectors of the sides; radius reaches each vertex.
• Inscribed circle (incircle) of a triangle: centre = incentre = intersection of the angle bisectors; radius = perpendicular distance to a side.
• Regular hexagon: each side equals the radius of the circumscribing circle (step the radius six times around the circle).
• Circumscribing/inscribing a circle in a regular hexagon; tangent at a point on a circle (perpendicular to the radius).

2. Formulas / Geometric Facts (quick reference)

• Circumcentre: equidistant from all vertices (intersection of perpendicular bisectors).
• Incentre: equidistant from all sides (intersection of angle bisectors).
• Length of a tangent from an external point at distance $d$ from the centre: $\sqrt{d^2-r^2}$.
• Regular hexagon: side $=$ circumradius; interior angle $=120^\circ$; it is made of 6 equilateral triangles.
• Tangent ⟂ radius at the point of contact.

3. Prerequisite Constructions/Formulas (in full)

• Perpendicular bisector of a segment (equidistant from both ends).
• Angle bisector (equidistant from both arms).
• Constructing 60°, 90°, 45°, 30°, 120° with a compass; copying a given angle.
• Angle sum: triangle $=180^\circ$, quadrilateral $=360^\circ$; interior angle of a regular n-gon $=\dfrac{(n-2)\times180^\circ}{n}$.
• Circle facts: circumference $2\pi r$, area $\pi r^2$ (often used alongside constructions).

4. Method Notes

• For an incircle: bisect two angles to get the incentre, then drop a perpendicular to a side for the radius.
• For a circumcircle: bisect two sides to get the circumcentre, then the radius is the distance to any vertex.
• Always keep construction arcs visible (don't erase) — marks earn credit.

5. Common Mistakes to Avoid

• Using angle bisectors for the circumcircle (those give the incircle) and vice versa.
• Erasing construction arcs (they must be shown).
• Forgetting a tangent is drawn perpendicular to the radius.
• Not using the radius as the side when constructing a hexagon.

6. Likely Exam Questions (with crisp answers)

1. How do you find the centre of the circumcircle? → Intersection of the perpendicular bisectors of the sides.
2. How do you find the centre of the incircle? → Intersection of the angle bisectors.
3. The circumcentre is equidistant from? → The three vertices.
4. The incentre is equidistant from? → The three sides.
5. Side of a regular hexagon vs its circumradius? → They are equal.
6. Interior angle of a regular hexagon? → 120°.
7. How are tangents from an external point constructed? → Using a circle on the line to the centre as diameter.
8. A tangent is drawn how, relative to the radius? → Perpendicular to it at the point of contact.
9. Interior angle of a regular n-gon? → $\frac{(n-2)180^\circ}{n}$.
10. Why keep construction arcs? → They show the method and earn marks.

Extended & Prerequisite Formulas (full)

• Circumcentre = ∩ of perpendicular bisectors (equidistant from vertices); incentre = ∩ of angle bisectors (equidistant from sides).
• Tangent length √(d²−r²); regular hexagon side = circumradius; interior∠ of regular n-gon $=\tfrac{(n-2)180^\circ}{n}$.
• Prereq: constructing 30/45/60/90/120°, perpendicular/angle bisectors; circle area & circumference.