An Angle Subtended By An Arc
2.8 Assumption: equal arcs on circles of equal radii subtend equal angles at the centre, and conversely.
The following results should be discussed and proofs given. Reproduction of memorised proofs will not be required.
Equal angles at the centre stand on equal chords. Converse.
The angle at the centre is twice the angle at the circumference subtended by the same arc.
The tangent to a circle is perpendicular to the radius drawn to the point of contact. Converse.
2.9 3 Unit students will be expected to be able to prove any of the following results using properties obtained in 2.3 or 2.8.
The perpendicular from the centre of a circle to a chord bisects the chord.
The line from the centre of a circle to the midpoint of a chord is perpendicular to the chord.
Equal chords in equal circles are equidistant from the centres.
Chords in a circle which are equidistant from the centre are equal.
Any three non-collinear points lie on a unique circle, whose centre is the point of concurrency of the perpendicular bisectors of the intervals joining the points.
Angles in the same segment are equal.
The angle in a semi-circle is a right angle.
Opposite angles of a cyclic quadrilateral are supplementary.
The exterior angle at a vertex of a cyclic quadrilateral equals the interior opposite angle.
If the opposite angles in a quadrilateral are supplementary then the quadrilateral is cyclic (also a test for four points to be concyclic).
If an interval subtends equal angles at two points on the same side of it then the end points of the interval and the two points are concyclic.
The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment.
Tangents to a circle from an external point are equal.
The products of the intercepts of two intersecting chords are equal.
The square of the length of the tangent from an external point is equal to the product of the intercepts of the secant passing through this point.
When circles touch, the line
The following results should be discussed and proofs given. Reproduction of memorised proofs will not be required.
Equal angles at the centre stand on equal chords. Converse.
The angle at the centre is twice the angle at the circumference subtended by the same arc.
The tangent to a circle is perpendicular to the radius drawn to the point of contact. Converse.
2.9 3 Unit students will be expected to be able to prove any of the following results using properties obtained in 2.3 or 2.8.
The perpendicular from the centre of a circle to a chord bisects the chord.
The line from the centre of a circle to the midpoint of a chord is perpendicular to the chord.
Equal chords in equal circles are equidistant from the centres.
Chords in a circle which are equidistant from the centre are equal.
Any three non-collinear points lie on a unique circle, whose centre is the point of concurrency of the perpendicular bisectors of the intervals joining the points.
Angles in the same segment are equal.
The angle in a semi-circle is a right angle.
Opposite angles of a cyclic quadrilateral are supplementary.
The exterior angle at a vertex of a cyclic quadrilateral equals the interior opposite angle.
If the opposite angles in a quadrilateral are supplementary then the quadrilateral is cyclic (also a test for four points to be concyclic).
If an interval subtends equal angles at two points on the same side of it then the end points of the interval and the two points are concyclic.
The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment.
Tangents to a circle from an external point are equal.
The products of the intercepts of two intersecting chords are equal.
The square of the length of the tangent from an external point is equal to the product of the intercepts of the secant passing through this point.
When circles touch, the line