Students:

Prove and apply angle and chord properties of circles (ACMMG272)

• identify and name parts of a circle (centre, radius, diameter, circumference, sector, arc, chord, secant, tangents, segment, semicircle)
• use terminology associated with angles in circles, e.g. subtend, standing on the same arc, angle at the centre, angle at the circumference, angle in a segment
• identify the arc on which an angle at the centre or circumference stands
• demonstrate that at any point on a circle there is a unique tangent to the circle, and that this tangent is perpendicular to the radius at the point of contact
• prove the following chord properties of circles:
  • chords of equal length in a circle subtend equal angles at the centre and are equidistant from the centre
  • the perpendicular from the centre of a circle to a chord bisects the chord; conversely, the line from the centre of a circle to the midpoint of a chord is perpendicular to the chord
  • the perpendicular bisector of a chord of a circle passes through the centre
  • given any three non-collinear points, the point of intersection of the perpendicular bisectors of any two sides of the triangle formed by the three points is the centre of the circle through all three points
  • when two circles intersect, the line joining their centres bisects their common chord at right angles
• use dynamic geometry software to investigate chord properties of circles (Problem Solving, Reasoning)
• prove the following angle properties of circles:
  • the angle at the centre of a circle is twice the angle at the circumference standing on the same arc
  • the angle in a semicircle is a right angle
  • angles at the circumference, standing on the same arc, are equal
  • the opposite angles of cyclic quadrilaterals are supplementary
  • an exterior angle at a vertex of a cyclic quadrilateral is equal to the interior opposite angle
• use dynamic geometry software to investigate angle properties of circles (Problem Solving, Reasoning)
• apply circle theorems to prove that the angle in a semicircle is a right angle (Problem Solving, Reasoning)
• apply chord and angle properties of circles to find unknown angles and lengths in diagrams

Prove and apply tangent and secant properties of circles

• prove the following tangent and secant properties of circles:
  • the two tangents drawn to a circle from an external point are equal in length
  • the angle between a tangent and a chord drawn to the point of contact is equal to the angle in the alternate segment
  • when two circles touch, their centres and the point of contact are collinear
  • the products of the intercepts of two intersecting chords of a circle are equal
  • the products of the intercepts of two intersecting secants to a circle from an external point are equal
  • the square of a tangent to a circle from an external point equals the product of the intercepts of any secants from the point
• use dynamic geometry software to investigate tangent and secant properties of circles (Problem Solving, Reasoning)
• apply tangent and secant properties of circles to find unknown angles and lengths in diagrams