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Circle Properties & Isosceles Triangles: Key Geometry Concepts

Learn about isosceles triangles, perpendicular bisectors, and essential circle properties involving angles, chords, and tangents. Identify key relationships to solve geometry problems efficiently.

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Circle Properties & Isosceles Triangles: Key Geometry Concepts

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  1. Isosceles triangles + perp. bisectors 1 Angle at center and at circumference 2 Angle in a semi circle 3 Angles in the same segment 4 Opposite angles in a cyclic quadrilateral 5 Rightangle between a tangent and radius 6 Tangential lines the same length 7 Alternate segment theorem 8

  2. Circle Property 1 Triangles formed using two radii will form an isosceles triangle. The perpendicular bisector of a chord passes through the the centre of the circle. Remember to spot isosceles triangles and perpendicular bisectors in circle diagrams.

  3. Circle Property 2 The angle subtended by an arc at the centre of a circle, is twice the angle subtended at the circumference. x x 2x The angle at the centre is half the angle on the circumference.

  4. Circle Property 3 Any angle subtended on the circumference of a semi- circle will be a right angle angle. x 1800 The angle in a semi-circle is a right angle.

  5. Circle Property 4 a a b a and b are both “subtended” by the same chord Bitesize

  6. Circle Property 5 Opposite angles in a cyclic quadrilateral add up to 1800. (A cyclic quadrilateral is a 4 sided shape with all four points on the circumference of a circle.) a x b Opposite angles in a cyclic quadrilateral add to 1800.

  7. Circle Property 6 Q The tangent to a circle is perpendicular to the radius drawn at the point of contact. OPQ = 900 P x O A tangent to a circle is at right angles to its radius.

  8. Circle Property 7 Q Two tangents drawn to a circle from the same point are equal in length. QP = QR QP = QR P x O R The tangents drawn from a point to a circle are equal in length.

  9. Circle Property 8 The angle between a tangent and a chord drawn at a point of contact is equal to any angle in the alternate segment. y x x z y The angle between a tangent and a chord is equal to the angle in the alternate segment.

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