This activity is to be used with pre-service mathematics teachers, in-service teachers in a project of professional development or high school students of Geometry. The main goal is to demonstrate how to use the TI-nspire technology to enhance the students capacity to elaborate and demonstrate conjectures.
The relationship between the lengths of the sides of a triangle inscribed in a circle, the Sine of the internal angles and the radius of the circle is studied.
Before the Activity
Install the file ExtendedLawofSines.tns in all the TI-handhelds. Students must have copy of the document LES-students.pdf or LES-students.doc. Students must be divided in groups of three.
During the Activity
Students will discuss with their group mates if they have previously heard about the Extended Law of Sines. If it is so, they will try to remember the statement and will write it on the space that is provided.
All students will turn to the page that contains the statement of the Law. After moving the vertices of the triangle on the circumference and seeing the different representations of the Extended Law of Sines they will verify that it holds, making the appropriate measurements as indicated (page 8). The demonstration of the Law is based on two theorems that have been seen previously:
1) The angle inscribed in a semicircle is a right angle.
2) Given B and C, two fixed points on a circle, for two points A and J on the circle The students will verify that these two properties always hold (page 11). The proof of the Extended Law of the Sines begins on page 13. Students must describe thoroughly this page in light of the two previous properties. Later, the students will give the reasons for the demonstration of the Law that is provided (page 14).
The professor must ask if this proof holds for all the cases and he (she) must observe that it is also necessary to prove that the Law holds in the case
After the Activity
1. the perpendicular bisectors of the sides of a triangle concur at a point
2. the point of concurrence of the perpendicular bisectors of the sides of a triangle is the center of the circle circumscribed to the triangle.