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Triangles & Circles

Congruence, similarity, angle in a circle, tangents, and power of a point

Key Concepts
  • Congruence and similarity criteria for triangles
  • Circle theorems: angles in the same segment, cyclic quadrilaterals
  • Power of a point relates secants and tangents
  • Standard centres: centroid, incentre, circumcentre, orthocentre
Important Formulae
Area of a triangle ½ · base · height = ½ ab sinC
Inradius / circumradius r = Area/s; R = abc/(4·Area)
Ptolemy (cyclic quad) AC · BD = AB · CD + AD · BC
Quick Tips
  • A tangent is always perpendicular to the radius at the point of contact.
  • Start most problems with angle chasing and look for cyclic quadrilaterals.
Sample Practice Questions

  1. The angle in a semicircle is:

    • 45°
    • 60°
    • 90°
    • 180°
    Show answer

    Answer: 90°

  2. In similar triangles, corresponding areas are in ratio:

    • Equal to sides ratio
    • Square of corresponding sides ratio
    • Cube of sides ratio
    • Same as perimeter ratio
    Show answer

    Answer: Square of corresponding sides ratio

  3. Two chords intersect inside a circle. If PA × PB = PC × PD, this is the:

    • Ptolemy's theorem
    • Ceva's theorem
    • Intersecting chords theorem (power of a point)
    • Menelaus's theorem
    Show answer

    Answer: Intersecting chords theorem (power of a point)

  4. Excircle opposite to vertex A has radius r_A = ?

    • Δ/s
    • Δ/(s-a)
    • Δ(s-a)/s
    • Δ/a
    Show answer

    Answer: Δ/(s-a)

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Practice Questions

Practise PRMO questions on Triangles & Circles. Answers are revealed after each question.

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Geometry