Constructions
About 7 min · Mathematical Reasoning · English
Time worked out from the lesson's 550 words at 150 words a minute, plus a minute for each question.
Constructions is a hands-on geometry topic that tests your ability to use only a compass and straight-edge (unmarked ruler) to draw precise geometric figures. In SOF IMO, you won't physically draw constructions during the exam, but you must understand the logical steps, the order of operations, and the geometric principles behind each construction. Questions may ask you to identify the correct sequence of steps, recognize which construction method applies, or determine properties of constructed figures.
Key ideas
- Compass and straight-edge only: All constructions must be done without measuring angles or lengths. The compass transfers distances; the straight-edge connects points. No protractor or marked ruler allowed.
- Arc intersections create key points: Most constructions rely on drawing arcs from two different centers. Where arcs intersect defines new points with specific geometric properties (equidistant from centers, for example).
- Angle bisector divides an angle into two equal parts: Constructed by drawing equal arcs from the angle vertex, then arcs from where those arcs cut the arms, creating an intersection that lies on the bisector.
- Perpendicular bisector of a segment: Every point on this line is equidistant from the segment's endpoints. Constructed by arcs of equal radius from both endpoints; their intersections determine the perpendicular bisector.
- Triangle construction uses congruence criteria: SSS (three sides), SAS (two sides and included angle), ASA (two angles and included side), RHS (right angle, hypotenuse, one side). Each criterion has a specific construction sequence.
Watch out for
- Changing compass width mid-construction → Arcs won't intersect properly or will create wrong distances. Fix: Set compass once per step and maintain that radius for all arcs in that step.
- Using too small a radius for perpendicular bisector → Arcs won't intersect if radius ≤ ½AB. Fix: Always open compass wider than half the segment length (visually estimate 60–70% of the segment).
- Forgetting to bisect first to get 30°, 45°, or other derived angles → Students try to construct 30° directly. Fix: 30° = bisect 60°; 45° = bisect 90°; 15° = bisect 30°. Build complex angles from basic ones.
Quick reference
- Angle bisector: equal arcs from vertex → equal arcs from arm intersections → join vertex to final intersection.
- Perpendicular bisector: arcs from both endpoints (radius > ½length) → join arc intersections.
- SSS triangle: draw base → arcs from both ends with given radii → join intersection to base ends.
- 60° angle: equilateral triangle property (compass = side length) gives 60° at each vertex.
- Tangent from external point P to circle O: draw PO → perpendicular bisector of PO → circle on midpoint → intersections are tangent points.
Check yourself: 3 questions
1. In the construction of the perpendicular bisector of a line segment AB, what is the minimum number of arcs that must be drawn using a compass?
2. A triangle ABC is to be constructed given that AB = 6 cm, angle A = 45 degrees, and AC = 5 cm. Which construction method should be used?
3. From an external point P, two tangents PA and PB are drawn to a circle with center O. If angle APB = 60 degrees, what is the measure of angle AOB?
This lesson is cut from the topic's note, not rewritten. Fact-checked on 21 Oct 2026: an AI-assisted check of this note against cited sources; the errors it found were corrected that day. Some errors may remain. The questions are from Shishya's validated question bank.
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