Project AMP Dr. Antonio R. Quesada – Director, Project AMP Inscribed Right Triangles Inscribed Right Triangles This lesson introduces students to the properties of inscribed right triangles. The properties are: 1. If a right triangle is inscribed in a circle, then its hypotenuse is a diameter of the circle. 2. If one side of a triangle inscribed in a circle is a diameter of the circle, then the triangle is a right triangle and the angle opposite the diameter is the right angle. Keywords: Right Triangles, Inscribed, Diameter, Hypotenuse Existing Knowledge These above properties are normally taught in a chapter concerning circles. Students should have a clear understanding of vocabulary concerning circles and triangles. NCTM Standards Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships. Learning Objectives Students will be able to identify the properties that exist when a right triangle is inscribed in a circle. Materials Cabri II or Geometer’s Sketchpad Procedure To introduce the lesson, the teacher may want to pose the question, “What is special about a right triangle inscribed in a circle?” If the students have previous experience with finding the circumcenter of a triangle, you may want to ask the question, “Given a right triangle, what is the easiest way to find the center of a circumscribed circle?” The teacher may assess the students based on questions throughout the lesson. Project AMP Dr. Antonio R. Quesada – Director, Project AMP Inscribed Right Triangles Team Members: ____________________ ____________________ File Name: ____________________ Activity Goal In this activity you will determine what properties exist when a right triangle is inscribed in a circle. Open a new page in Cabri II and follow the instructions below. Laboratory One 1. Construct a circle and label the center C. (circle tool) 2. Inscribe right DABD in circle C using the instructions below. a. Construct chord AB . (segment tool) b. Construct a line through B that is perpendicular to AB . (perpendicular line tool) c. Label the intersection of the circle and the line as D. (intersection point tool) d. Construct AD. (segment tool) e. Measure ÐABD to show that it is a right angle. (angle measure tool) 3. Are A-C-D collinear? ____ (collinear tool) Project AMP Dr. Antonio R. Quesada – Director, Project AMP 4. What is the relationship of AD to the circle? Explain. _____________________________________________________________________ 5. What is the relationship of AD to DABD ? Why? _______________________ ___________________________________________________________________________ 6. What is the relationship between the hypotenuse and the diameter of the circle? ___________________________________________________________________________ 7. Grab and move point B about the circle. 8. Does the measure of DABD change? _____ 9. Does the position of AD change? _____ 10. What can you conclude about any right triangle inscribed in a circle? ___________________ ___________________________________________________________________________ Project AMP Dr. Antonio R. Quesada – Director, Project AMP Laboratory Two 1. Construct a circle with center C. (circle tool) 2. Construct line m through point C. (line tool) 3. Label the intersections of line m and circle C as points A and B. (intersection point tool) 4. Construct AB and then hide line m. (segment tool and hide tool) 5. What is AB in relationship to the circle? _________________________________ 6. Put a point D on the circle and then construct the inscribed DABD . (point tool and triangle tool) 7. What is the measure of ÐADB? ____________________________ (measurement tool) 8. What kind of triangle is DADB ? ______________________________ 9. What is the hypotenuse of DADB ? ____________________________ 10. Grab and move point D about the circle. 11. What happens to the measure of ÐADB? ____________________________ 12. What can you conclude about any inscribed triangle which has one side as the diameter of the circle? _____________________________________________________________________ ___________________________________________________________________________ Project AMP Dr. Antonio R. Quesada – Director, Project AMP 13. Lab 2 is the converse of lab 1. Summarize each lab in your own words. Lab 1: ________________________________________________________________________ ______________________________________________________________________________ Lab 2: ________________________________________________________________________ ______________________________________________________________________________ Extension #1 Given a right triangle, what is the easiest way to find the center of a circumscribed circle? ______________________________________________________________________________ ______________________________________________________________________________ Extension #2 When you go to a concert, you want to be close to the stage, but you don’t want to have to move your eyes too much to see all the dancers and musicians on each end. In the diagram, the measure of ÐXYZ is called your viewing angle. You decide that the middle of the sixth row has the best viewing angle. If someone is sitting there, explain where else you could sit to have the same viewing angle. If necessary, make a diagram or use Cabri II. ______________________________________________________________________________ ______________________________________________________________________________ .