It is expected that students will:

• use technology with dynamic geometry software and measurement to confirm and apply the following properties:

- the perpendicular from the centre of a circle to a chord bisects the chord
- the measure of the central angle is equal to twice the measure of the inscribed angle subtended by the same arc
- the inscribed angles subtended by the same arc are congruent

Determine the measure of angle x. Justify your assertions.

- the angle inscribed in a semicircle is a right angle
- the opposite angles of a cyclic quadrilateral are supplementary
- a tangent to a circle is perpendicular to the radius at the point of tangency
- the tangent segments to a circle, from any external point, are congruent
- the angle between a tangent and a chord is equal to the inscribed angle on the opposite side of the chord
- the sum of the interior angles of an n-sided polygon is (2n - 4) right angles

Determine the measure of angle x. Justify your assertions.

Draw a semicircle with diameter AB. Draw an angle, ACB, with C being any point on the semicircle. What is the measure of angle ACB? Repeat for two other points C', and C", on the semicircle. What pattern emerges?

Determine the measure of , where E is the centre of the circle.

How far from the inside corner of the shelf, A, is the centre C of the plate, if the plate has a diameter of 20 cm?

Find the length of the side of the square if the diameter of the semicircle is 20 cm. Justify your assertions.

The perimeter of the isosceles triangle ABC, with AC = BC, is 54 cm. If AD = 5 cm, and D, E and F are points of tangency, find the length of BC.

Determine the measure of and .

A circular hole 30 cm in diameter is cut in a sheet of metal and a spherical globe 34 cm in diameter is set into the hole. How far below the top surface of the metal sheet will the sphere extend?

• prove the following general properties, using established concepts and theorems:

a) For what values of c does the line y = c touch the circle
b) Use the result from part a) to show that the tangent to a circle is perpendicular to the radius at the point of tangency.

- the perpendicular bisector of a chord contains the centre of the circle
- the measure of the central angle is equal to twice the measure of the inscribed angle subtended by the same arc (for the case when the centre of the circle is in the interior of the inscribed angle)

Show that the angle inscribed in a semicircle is a right angle.

- the inscribed angles subtended by the same arc are congruent
- the angle inscribed in a semicircle is a right angle
- the opposite angles of a cyclic quadrilateral are supplementary
- a tangent to a circle is perpendicular to the radius at the point of tangency
- the tangent segments to a circle from any external point are congruent
- the angle between a tangent and a chord is equal to the inscribed angle on the opposite side of the chord
- the sum of the interior angles of an n-sided polygon is (2n - 4) right angles.

*The chord AB is one side of a regular polygon of n sides. The polygon is inscribed in a circle. If D is any other vertex of the polygon, prove that the magnitude of angle ADB is .

Explore the relationships between the number of sides of a regular polygon and the sum of the internal angles.

• solve problems, using a variety of circle properties, and justify the solution strategy used

If diameter CD is perpendicular to chord AB at E, prove that triangle ABC is isosceles.

Show that BC bisects angle ECA, if AC is the diameter of the circle, CE is perpendicular to DF and DF is a tangent at B.

Determine the measure of and
. Provide a reason for each step in the solution strategy.

*A chain on a bicycle connects two gear wheels of diameters 9 cm and 19 cm respectively. The centres of the gear wheels are 87 cm apart. Find the minimum length of the chain.