SOF IMO · Bite-size lessons

Circles

About 6 min · Mathematical Reasoning · English

Time worked out from the lesson's 430 words at 150 words a minute, plus a minute for each question.

Circle geometry is part of the SOF IMO Mathematical Reasoning section, testing your understanding of fundamental theorems and their applications. Questions may combine multiple properties — for instance, a chord problem that also requires angle calculations or a cyclic quadrilateral merged with tangent properties. This topic rewards students who can visualize configurations quickly and recall theorems precisely.

Key ideas

• Equal chords equidistant from center: Chords equal in length lie at equal perpendicular distances from the circle's center, and conversely, chords equidistant from the center are equal in length.
• Perpendicular from center bisects chord: A line from the center perpendicular to a chord always bisects that chord; this creates right triangles useful for applying Pythagoras.
• Angles subtended by an arc: An arc subtends twice the angle at the center compared to any point on the remaining part of the circle (angle at center = 2 × angle at circumference).
• Angles in the same segment: All angles subtended by the same arc at different points on the circle (in the same segment) are equal.
• Angle in a semicircle: Any angle subtended by a diameter at a point on the circle is a right angle (90°).

Watch out for

• Assuming any chord is bisected by any line from center → Only a perpendicular from the center bisects a chord; slanted lines do not.
• Confusing angle at center with angle at circumference → Remember the 2:1 ratio: the central angle is always double the inscribed angle for the same arc.
• Forgetting the "same segment" condition → Angles are equal only if they're in the same segment; angles on opposite sides of a chord subtend different arcs.

Quick reference

• Perpendicular from center bisects any chord.
• Angle at center = 2 × angle at circumference (same arc).
• Angle in semicircle = 90° always.
• Cyclic quadrilateral: opposite angles sum to 180°.
• Tangent ⊥ radius at point of contact; equal tangents from external point.

Check yourself: 3 questions

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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