Grade Level: High School Geometry

Learning Objectives:

After completing this unit, students shuld be able to demonstrate their knowledge and understanding of geometric means and right triangles by:

• Finding the relationships among the lengths of the sides of a right triangle and the altitude to the hypotenuse.
• Utilizing a dynamic geometric software package to investigate the relationships stated in objective one.
• Determining the geometric mean between two numbers.
• Stating and applying the relationships that exist when the altitude is drawn to the hypotenuse of a right triangle.

This lesson attains many of the goals set forth in the Standards 1, 2, 3, 4, 7, and 8 (Mathematics as Problem Solving, Mathematics as Communication, Mathematics as Reasoning, and Mathematical Connections, and the Geometry Standards respectively.)

In accordance with the Standards, the teacher's primary role is to introduce students to various attributes of right triangles and the fundamental concepts of geometric means. His/her primary role will be to ask for justification of student answers. Further, the teacher should assist students in the utilization of the geometry package which the students will use to form and test their various conjectures. Likewise, the teacher should encourage the students to work cooperatively to develop and test their conjectures and explanations.

• Students should know the parts of a right triangle: the two legs and hypotenuse.
• Students should be know the difference among altitude, median, and perpendicular bisector.
• Students need to be able to recognize similar triangles and their proportions between them.
• Students should have a "working" knowledge of a dynamic geometry program such as Geometer's Sketchpad or Cabri Geometry.

Warm Up:

To introduce the unit, the teacher could begin with a discussion of the differences between arithmetic and geometric means. After calculating each type of mean, the teacher might want to review the various parts of a right triangle to insure that all students are utilizing their correct mathematical vocabulary skills.

The Lesson:

Students often have a difficult a difficult time with the geometric means and right triangles. One of the many goals of this lesson is to allow the students to use a dynamic geometry program such as Geometer's Sketchpad or Cabri Geometry to investigate the various relationships which occur among the lengths of different pieces of the right triangles. Depending on the availability of computers, this lesson could be presented either (1) as a demonstration/discussion/investigation lead by either the teacher or one of the students or (2) as an investigation by either small groups or each individual student. However, depending on the amount of time for presentation/investigation of the lesson, the entire sketch may be created and investigated or the sketches can be downloaded by clicking on each of the following links Altitude and Hypotenuse of a Right Triangle and Legs and Hypotenuse of a Right Triangle. Note: All investigations can be performed using the same sketch.

Possible Discusion Questions or Investigation Ideas:

1. How many triangles exist in the diagram? Name each of the them.
2. Compare the interior angle measures of the three triangles.
3. List the congruent angles.
4. Based on the above information, what is true about the three triangles? Write a statement showing the relationship among the three triangles.
5. After measuring the length of each segment in the sketch, what conclusions can you make regarding the triangles listed in Question 1? Verify your conclusions with the appropriate theorem or algebraic rationale.
6. Based on your conclusions above and the lengths of each of the segments, what can you conjecture about the length of one specific segment and the geometric mean of the segments of the hypotenuse? What is that specific segment called?
7. Two other geometric means exist within the triangles. What conjectures can you form and test to discover them? Write your conclusions as complete statements.

Using Geometer's Sketchpad or some other dynamic geoemtry program, calculate the product of the segments of the hypotenuse. Select the measures of the altitude and this product and choose Plot As (x,y) from the pull-down Graph menu. Construct the locus of this plotted point and the vertex of the large right triangle. Describe the resulting graph.