Learning Resources
Overview
Materials
Throughout this unit at various times you will need the following materials, be sure to bring them to class with you each day as you are doing this unit.
- a ruler longer than 18 cm
- a set of compasses to draw a circle
- a pair of scissors
- graph paper
- a protractor
Overview
In this unit you will use skills and concepts from transformational, Euclidean, and coordinate geometry .
The chord properties are investigated using several different approaches, and they are related to aspects of the design of a sports complex. The concept of a converse is introduced, and the "iff" notation is explained. The Pythagorean theorem is reviewed in context. Congruent triangles are reviewed and applied, as are transformational and Euclidean proofs.
You will examine the slopes of parallel and perpendicular lines, using coordinate geometry as the context of the sports complex is continued. The distance between two points is investigated for horizontal and vertical lines, which leads to the development of the distance formula for oblique lines. The formula for the midpoint of a line is introduced and used in context. You will check the chord properties using coordinate geometry.
You will work with major and minor arcs, and semi-circles. You will discover relationships between the measures of arcs and the central angles and inscribed angles subtended by the arcs. You will be introduced to the concept of arc measure using portable rings for a circus. You will examine properties of tangents to a circle and use various methods to prove these properties. You will then use this knowledge to solve practical problems.
You will graph circles with centres not at the origin. You will examine various transformations of equations of circles and the effect of these transformations on the graph. You will use graphing technology to further understand equations of circles. You will examine the ellipse as a transformation of a circle.
Finally you will explore how to use trigonometric functions to describe any point on a circle centred at the origin. With the aid of graphing calculators, you will use trigonometric parametric equations to graph circles whose centres are not necessarily at the origin. As well, the arc measure will also be related to the corresponding central angle measured in radians.
Recall the prior knowledge you have with respect to circles even though you have not studied these concepts for some time. If there are ideas you just do not remember, ask some one for help; there are many circle properties and geometric properties referred to in this unit that you are expected to already know.
This unit should take approximately 30 hours to complete.