Page 136 - The Ontario Curriculum, Grades 11 and 12: Mathematics, 2007

Grade 12, College Preparation
A 5 km
5 km D
BC
4 km
45°
45°
10 ft
50 ft
20 ft
1.0 cm 1.2 cm 1.0 cm
1.0 cm 1.0 cm
130°
3. Solving Problems Involving Circle Properties
THE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics
By the end of this course, students will:
3.1 recognize and describe (i.e., using diagrams and words) arcs, tangents, secants, chords, segments, sectors, central angles, and inscribed angles of circles, and some of their real-world applications (e.g., construction of a medicine wheel)
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reason that sewer covers are round is to pre- vent them from falling into the sewer during removal and replacement)
Sample problem: Explain why rectangular prisms are often used for packaging.
2.2 perform required conversions between the imperial system and the metric system using a variety of tools (e.g., tables, calculators, online conversion tools), as necessary within applications
2.3 solve problems involving the areas of rect- angles, parallelograms, trapezoids, triangles, and circles, and of related composite shapes, in situations arising from real-world applications
Sample problem: Your company supplies circular cover plates for pipes. How many plates with a 1-ft radius can be made from a 4-ft by 8-ft sheet of stainless steel? What percentage of the steel will be available for recycling?
2.4 solve problems involving the volumes and surface areas of spheres, right prisms, and cylinders, and of related composite figures, in situations arising from real-world applications
Sample problem: For the small factory shown in the following diagram, design specifica- tions require that the air be exchanged every 30 min. Would a ventilation system that exchanges air at a rate of 400 ft3/min satisfy the specifications? Explain.
3.2 determine the length of an arc and the area of a sector or segment of a circle, and solve relat- ed problems
Sample problem: A circular lake has a diame- ter of 4 km. Points A and D are on opposite sides of the lake and lie on a straight line through the centre of the lake, with each point 5 km from the centre. In the route ABCD, AB and CD are tangents to the lake and BC is an arc along the shore of the lake. How long is this route?
3.3 determine, through investigation using a vari- ety of tools (e.g., dynamic geometry software), properties of the circle associated with chords, central angles, inscribed angles, and tangents (e.g., equal chords or equal arcs subtend equal central angles and equal inscribed angles; a radius is perpendicular to a tangent at the point of tangency defined by the radius, and to a chord that the radius bisects)
Sample problem: Investigate, using dynamic geometry software, the relationship between the lengths of two tangents drawn to a circle from a point outside the circle.
3.4 solve problems involving properties of circles, including problems arising from real-world applications
Sample problem: A cylindrical metal rod with a diameter of 1.2 cm is supported by a wood- en block, as shown in the following diagram. Determine the distance from the top of the block to the top of the rod.