Inscribed+and+Circumscribed+Circles

**G.C.3**
 * Circles Unit**
 * Goal:** Students will be able to construct an inscribed and circumscribed circle.


 * 1) Students will be able to distinguish between inscribed and circumscribed circles.
 * 2) Students will understand which special segments of a triangle are used to construct inscribed and circumscribed circles.


 * __Integrate and mention Practice Standards throughout the learning cycles:__**
 * 1) Make sense of problems and persevere in solving them.
 * 2) Reason abstractly and quantitatively.
 * 3) Construct viable arguments and critique the reasoning of others.
 * 4) Model with mathematics.
 * 5) Use appropriate tools strategically.
 * 6) Attend to precision.
 * 7) Look for and make use of structure.
 * 8) Look for and express regularity in repeated reasoning.


 * __Content 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.

Goal: Construct the inscribed and circumscribed circles of a triangle.

Objectives:
 * Inscribe a circle in a triangle
 * Circumscribe a circle about a triangle


 * __Develop Understanding__**


 * Task #1:** __Pivot Line__

//Students will need to have access to rulers, compasses, patty paper.//

A farmer in Moroni has purchased a pivot line to water his triangular piece of property in which he grows alfalfa. Where should the pivot line be located so that he maximizes his field space?

Question 1: Can you determine where your pivot point should be without the use of any tools?

Question 2: The pivot point you found is the center of your circle. Determine if this point is an orthicenter, circumcenter, incenter or centroid. Use this information to construct an inscribed circle on your patty paper. How close your estimate?


 * Launch:** Introduce the task as a challenge and let the students read the directions then check for understanding of the task.


 * Explore:** Questions 1 & 2 should be on separate pieces of paper. Give students time to explore question 1 without teacher involvement. A discussion of question 1 should occur before you hand out question 2. In exploration of question 2, important vocabulary must be addressed. Patty paper is a useful tool for question 2.


 * Discuss:** Select students to share their work with the class. Students should begin to use vocabulary in their explanations (i.e: median, altitude, perpendicular bisector, angle bisectors). A review of each construction may be needed.


 * __Solidify Understanding__**


 * Task #2:** Selecting a Hospital Location

//Students will need to have access to rulers, compasses, patty paper.//

Tooele County would like to build a hospital centrally located between Grantsville, Stansbury, and Tooele. Where should the hospital be located to be equally located between all three cities?


 * Launch:** Introduce the task as a challenge and let the students read the directions then check for understanding of the task.


 * Explore:** Students will make connections from task 1 to discover the circumcenter of the triangle. Using a map of these cities would be useful and interesting.


 * Discuss:** Select students to share their work with the class. Students should begin to use vocabulary in their explanations (i.e: median, altitude, perpendicular bisector, angle bisectors). A review of each construction may be needed.


 * __Practice Understanding__**

Give the students a copy of a map and have the students construct an inscribed and circumscribed circle connecting any three noncollinear cities and explore the relationship of their centers. Explore how the circles are different for different types of triangles (ie: obtuse, right, acute). Will the centers ever be the same?