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Draw and label on a circle:Centre Radius Diameter Circumference Chord Tangent Arc Sector (major/minor) Segment (major/minor)
Circle Fact 1. Isosceles TriangleAny triangle AOB with A & B on the circumference and O at the centre of a circle is isosceles. r a a r a
Circle Fact 2. Tangent and RadiusThe tangent to a circle is perpendicular to the radius at the point of contact.
Circle Fact 3. Two TangentsThe 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 AngleThe angle subtended by an arc at the centre of a circle is twice the angle subtended at the circumference.
Circle Theorem 2: SemicircleThe angle in a semicircle is a right angle.
Circle Theorem 3: Segment AnglesAngles in the same segment are equal.
Circle Theorem 4: Cyclic QuadrilateralThe sum of the opposite angles of a cyclic quadrilateral is 180o.
Circle Theorem 5: Alternate SegmentThe 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 AngleProof The angle subtended by an arc at the centre of a circle is twice the angle subtended at the circumference.
b b 2a + 2b = 2(a + b) b 180 – 2a a a
Circle Theorem 2: SemicircleProof The angle in a semicircle is a right angle.
b 360 – 2(a + b) = 180 180 = 2(a + b) 180 – 2b 90 = (a + b) 180 – 2a b a a
Circle Theorem 3: Segment AnglesProof Angles in the same segment are equal.
a a 2a
Circle Theorem 4: Cyclic QuadrilateralProof The sum of the opposite angles of a cyclic quadrilateral is 180o.
a 2a + 2b = 360 2(a + b) = 360 2b a + b = 180 2a b
Circle Theorem 5: Alternate SegmentProof The angle between a chord and the tangent at the point of contact is equal to the angle in the alternate segment.
90 - a a 180 – 2(90 – a) 2a 180 – a 90 - a a
1 2 3 4 5 Double Angle Semicircle Segment Angles Cyclic QuadrilateralAlternate Segment
Circle Theorem Remember to look for “basics” Angles in a triangle sum to Angles on a line sum to Isosceles triangles (radius) Angles about.
Circle Theorems The angle at the centre is twice the angle at the circumference for angles which stand on the same arc.
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