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Circle theorems

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Thabang Nombewu
Thabang Nombewu

Euclidean Geometry is the integral part of High school Math. The concept needs to learned accurately!

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Draw and label on a circle: Centre Radius Diameter Circumference Chord Tangent Arc Sector (major/minor) Segment (major/minor)
a a 180 - 2a r r Circle Fact 1. Isosceles Triangle Any triangle AOB with A & B on the circumference and O at the centre of a circle is isosceles.
Circle Fact 2. Tangent and Radius The tangent to a circle is perpendicular to the radius at the point of contact.
Circle Fact 3. Two Tangents The triangle produced by two crossing tangents is isosceles.
Circle Fact 4. Chords If a radius bisects a chord, it does so at right angles, and if a radius cuts a chord at right angles, it bisects it.
Circle Theorem 1: Double Angle The angle subtended by an arc at the centre of a circle is twice the angle subtended at the circumference.
Circle Theorem 2: Semicircle The angle in a semicircle is a right angle.
Circle Theorem 3: Segment Angles Angles in the same segment are equal.
Circle Theorem 4: Cyclic Quadrilateral The sum of the opposite angles of a cyclic quadrilateral is 180o .
Circle Theorem 5: Alternate Segment The angle between a chord and the tangent at the point of contact is equal to the angle in the alternate segment.
Circle Theorem 1: Double Angle The angle subtended by an arc at the centre of a circle is twice the angle subtended at the circumference.
a a b b 180 - 2b 180 – 2a 2a + 2b = 2(a + b)
Circle Theorem 2: Semicircle The angle in a semicircle is a right angle.
180 – 2a 180 – 2b a a b b 360 – 2(a + b) = 180 180 = 2(a + b) 90 = (a + b)
Circle Theorem 3: Segment Angles Angles in the same segment are equal.
a a 2a
Circle Theorem 4: Cyclic Quadrilateral The sum of the opposite angles of a cyclic quadrilateral is 180o .
a b 2a 2b 2a + 2b = 360 2(a + b) = 360 a + b = 180
Circle Theorem 5: Alternate Segment The angle between a chord and the tangent at the point of contact is equal to the angle in the alternate segment.
a 90 - a 90 - a 180 – 2(90 – a) 180 – 180 + 2a 2a a
Double Angle Semicircle Segment Angles Cyclic Quadrilateral Alternate Segment

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Circle theorems

  • 1. Draw and label on a circle: Centre Radius Diameter Circumference Chord Tangent Arc Sector (major/minor) Segment (major/minor)
  • 2. a a 180 - 2a r r Circle Fact 1. Isosceles Triangle Any triangle AOB with A & B on the circumference and O at the centre of a circle is isosceles.
  • 3. Circle Fact 2. Tangent and Radius The tangent to a circle is perpendicular to the radius at the point of contact.
  • 4. Circle Fact 3. Two Tangents The triangle produced by two crossing tangents is isosceles.
  • 5. Circle Fact 4. Chords If a radius bisects a chord, it does so at right angles, and if a radius cuts a chord at right angles, it bisects it.
  • 6. Circle Theorem 1: Double Angle The angle subtended by an arc at the centre of a circle is twice the angle subtended at the circumference.
  • 7. Circle Theorem 2: Semicircle The angle in a semicircle is a right angle.
  • 8. Circle Theorem 3: Segment Angles Angles in the same segment are equal.
  • 9. Circle Theorem 4: Cyclic Quadrilateral The sum of the opposite angles of a cyclic quadrilateral is 180o .
  • 10. Circle Theorem 5: Alternate Segment The angle between a chord and the tangent at the point of contact is equal to the angle in the alternate segment.
  • 11. Circle Theorem 1: Double Angle The angle subtended by an arc at the centre of a circle is twice the angle subtended at the circumference.
  • 12. a a b b 180 - 2b 180 – 2a 2a + 2b = 2(a + b)
  • 13. Circle Theorem 2: Semicircle The angle in a semicircle is a right angle.
  • 14. 180 – 2a 180 – 2b a a b b 360 – 2(a + b) = 180 180 = 2(a + b) 90 = (a + b)
  • 15. Circle Theorem 3: Segment Angles Angles in the same segment are equal.
  • 17. Circle Theorem 4: Cyclic Quadrilateral The sum of the opposite angles of a cyclic quadrilateral is 180o .
  • 18. a b 2a 2b 2a + 2b = 360 2(a + b) = 360 a + b = 180
  • 19. Circle Theorem 5: Alternate Segment The angle between a chord and the tangent at the point of contact is equal to the angle in the alternate segment.
  • 20. a 90 - a 90 - a 180 – 2(90 – a) 180 – 180 + 2a 2a a
  • 21. Double Angle Semicircle Segment Angles Cyclic Quadrilateral Alternate Segment