Secondary Mathematics II
Educational Links

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).

• 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.
• Mutually Tangent Circles
  This is a challenging task which requires students to carefully divide up the picture into different pieces for which the area is known.
• 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.
• Right triangles inscribed in circles I
  This task provides a good opportunity to use isosceles triangles and their properties to show an interesting and important result about triangles inscribed in a circle with one side of the triangle a diameter: the fact that these triangles are always right triangles is often referred to as Thales' theorem
• Right triangles inscribed in circles II
  This task is designed to address the standard "Identify and describe relationships among inscribed angles, radii, and chords."
• Sectors of Circles
  This lesson unit is intended to help you assess how well students are able to solve problems involving area and arc length of a sector of a circle using radians. It assumes familiarity with radians and should not be treated as an introduction to the topic.
• Similar circles
  The goal of this task is to work on showing that all circles are similar using these two different methods, the first visual and the second algebraic.
• Solving Problems with Circles and Triangles
  This lesson unit is intended to help educators assess how well students are able to use geometric properties to solve problems
• Student Task: Circles in Triangles
  In this task, the students have to find the radius of circles inscribed in various sizes of right triangle.
• Student Task: Temple Geometry
  During the Edo period (1603-1867) of Japanese history, geometrical puzzles were hung in the holy temples as offerings to the gods and as challenges to worshippers. Here is one such problem for students to investigate.
• Tangent Lines and the Radius of a Circle
  This task presents a foundational result in geometry, presented with deliberately sparse guidance in order to allow a wide variety of approaches.
• Tangent to a circle from a point
  This task is designed to allow students to construct a tangent line from a point outside a given circle to the circle.
• Two Wheels and a Belt
  This task combines two skills from domain G-C: making use of the relationship between a tangent segment to a circle and the radius touching that tangent segment (G-C.2), and computing lengths of circular arcs given the radii and central angles (G-C.5).

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