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Secondary Mathematics II

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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. -
GEOMETRY - Circles (G.C) - Sec Math II Core Guide

The Utah State Board of Education (USBE) and educators around the state of Utah developed these guides for the Secondary Mathematics II - Circles (G.C). -
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).

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 - Joleigh Honey and see the Mathematics - Secondary website. For general questions about Utah's Core Standards contact the Director - Jennifer Throndsen . These materials have been produced by and for the teachers of the State of Utah. Copies of these materials may be freely reproduced for teacher and classroom use. When distributing these materials, credit should be given to Utah State Board of Education. These materials may not be published, in whole or part, or in any other format, without the written permission of the Utah State Board of Education, 250 East 500 South, PO Box 144200, Salt Lake City, Utah 84114-4200.