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Topic 3: Circle and geometric proofs

Circle properties and their proofs

6 hours

• Prove the circle properties — the angle at the centre subtended by an arc of a circle is twice the angle at the circumference subtended by the same arc; an angle in a semicircle is a right angle; angles at the circumference of a circle subtended by the same arc are equal; the alternate segment theorem; the opposite angles of a cyclic quadrilateral are supplementary and its converse; a tangent drawn to a circle is perpendicular to the radius at the point of contact and its converse. 2 interactives
• Solve problems finding unknown angles and lengths and prove further results using the circle properties listed above.

Geometric proofs using vectors

6 hours

• Prove the diagonals of a parallelogram meet at right angles if and only if it is a rhombus. 1 interactive
• Prove midpoints of the sides of a quadrilateral join to form a parallelogram. 1 interactive
• Prove the sum of the squares of the lengths of a parallelogram’s diagonals is equal to the sum of the squares of the lengths of the sides. 1 interactive
• Prove an angle in a semicircle is a right angle. 1 interactive