Circle theorems - Part 3

Circle theorems - Part 3

Angles in a Cyclic Quadrilateral

  • A cyclic quadrilateral is a four-sided figure in which all vertices lie on a circle’s circumference.
  • The sum of the opposite angles of a cyclic quadrilateral is always 180°. This is referred to as the opposite angles theorem in circle theorems.
  • Therefore, if you know three angles of a cyclic quadrilateral, you can find the fourth by subtracting the known angles from 360°.

The Alternate Segment Theorem

  • The Alternate Segment Theorem applies to triangles inscribed within a circle.
  • It states that the angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment.
  • In simpler terms, the base angles in an isosceles triangle inscribed in a circle are equal to the angle subtended by the base of the triangle to the circle.

Angles Subtended by the Same Arc

  • When two angles are subtended by the same arc (or chord) on a circle, they are equal. This is commonly called the Equal Chord Theorem in circle theorems.
  • It is essential to remember that the angles in question must be subtended to the same segment for this rule to apply.
  • For example, the angles subtended by the chord AB at the circumference will be equal, but the angle subtended by chord AB at the centre will be double.

Tangent and Diameter Perpendicularity

  • A line drawn from the centre of a circle perpendicular to a chord divides the chord into two equal parts. Similarly, the perpendicular from the centre to a chord bisects the chord.
  • A tangent drawn to a circle is always perpendicular to the radius from the point of tangency.
  • This theorem is especially useful in problems where the radius, tangent, and angle properties of a circle are to be exploited.

Remember these theorems while solving mathematical problems revolving around circles, as they provide crucial links and simplify complex problems significantly. Circle theorems form an integral part of geometry applications, hence the development of a sound understanding is essential.