 Secondary Mathematics II

Strand: GEOMETRY - Circles (G.C)

Understand and apply theorems about circles (Standard G.C.1�4). Find arc lengths and areas of sectors of circles. Use this as a basis for introducing the radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course (Standard G.C.5).

Standard G.C.3

Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

• Circumcenter of a triangle
This task shows that the three perpendicular bisectors of the sides of a triangle all meet in a point, using the characterization of the perpendicular bisector of a line segment as the set of points equidistant from the two ends of the segment.
• Circumscribed Triangles
The goal of this task is to study where a circumscribed triangle can meet a given circle.
• Inscribing a circle in a triangle I
This task shows how to inscribe a circle in a triangle using angle bisectors.
• Inscribing a circle in a triangle II
This task focuses on a remarkable fact which comes out of the construction of the inscribed circle in a triangle: the angle bisectors of the three angles of triangle ABC all meet in a point.
• Inscribing a triangle in a circle
This problem introduces the circumcenter of a triangle and shows how it can be used to inscribe the triangle in a circle. It also shows that there cannot be more than one circumcenter.
• Inscribing and Circumscribing Right Triangles
This lesson unit is intended to help educators assess how well students are able to use geometric properties to solve problems.
• Locating Warehouse
This task can be implemented in a variety of ways. For a class with previous exposure to the incenter or angle bisectors, part (a) could be a quick exercise in geometric constructions,. Alternatively, this could be part of a full introduction to angle bisectors, culminating in a full proof that the three angle bisectors are concurrent, an essentially complete proof of which is found in the solution below.
• Module 7: Circles from a Geometric Perspective - Student Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 7, Circles: A Geometric Perspective, is composed of four learning cycles. In the first learning cycle, students use rotations and perpendicular bisectors to find the center of a circle. The second learning cycle in Module 7 builds on the circle relationships that students have learned so far in the module to develop a formula for the perimeter and area of a regular polygon. The third learning cycle addresses relationships among central angles, radii, arcs, and sectors. Students calculate arc length and the area of a sector. The final learning cycle in Module 7 is an intuitive approach to volume of prisms, pyramids, and cylinders.
• Module 7: Circles from a Geometric Perspective - Teacher Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 7, Circles: A Geometric Perspective, is composed of four learning cycles. In the first learning cycle, students use rotations and perpendicular bisectors to find the center of a circle. The second learning cycle in Module 7 builds on the circle relationships that students have learned so far in the module to develop a formula for the perimeter and area of a regular polygon. The third learning cycle addresses relationships among central angles, radii, arcs, and sectors. Students calculate arc length and the area of a sector. The final learning cycle in Module 7 is an intuitive approach to volume of prisms, pyramids, and cylinders.
• Opposite Angles in a Cyclic Quadrilateral
The goal of this task is to show that opposite angles in a cyclic quadrilateral are supplementary.
• Placing a Fire Hydrant
This task can be implemented in a variety of ways. For a class with previous exposure to properties of perpendicular bisectors, part (a) could be a quick exercise in geometric constructions, and an application of the result. Alternatively, this could be part of an introduction to perpendicular bisectors, culminating in a full proof that the three perpendicular bisectors are concurrent at the circumcenter of the triangle.
• Student Task: Circles in Triangles
In this task, the students have to find the radius of circles inscribed in various sizes of right triangle. http://www.uen.org - in partnership with Utah State Board of Education (USBE) and Utah System of Higher Education (USHE).  Send questions or comments to USBE Specialist - Lindsey Henderson and see the Mathematics - Secondary website. For general questions about Utah's Core Standards contact the Director - Jennifer Throndsen.