Project AMP Dr. Antonio Quesada – Director, Project AMP
Exploring Geometric Mean
Lesson Summary: The students will explore the Geometric Mean through the use of Cabrii II software or TI – 92 Calculators and inquiry based activities.
Keywords: Geometric Mean, Ratios
NCTM Standards: 1. Analyze characteristics and properties of two-dimensional geometric shapes and develop mathematical arguments about geometric relationships (p.310). 2. Apply appropriate techniques, tools, and formulas to determine measurements (p.322).
Learning Objectives: 1. Students will be able to construct a right triangle with an altitude drawn from the right angle. 2. Students will be able to use the geometric mean to find missing measures for parts of a right triangle.
Materials Needed: 1. Cabri II Geometry Software 2. Pencil, Paper 3. Lab Handout
Procedures/Review: 1. Attention Grabber: Begin by asking the students, “What are some ways we can find missing lengths of a right triangle?” (The students may already know how to apply the Pythagorean Theorem, or the formulas for special right triangles, or maybe even how to use trigonometric ratios). The answer you will be showing them is how to use the geometric mean. 2. Students can be grouped in teams of two or, if enough computers are available, they may work individually. 3. Assessment will be based on the students’ completion of the lab worksheet. 4. A review of ratios and proportions may be necessary prior to instruction.
Project AMP Dr. Antonio Quesada – Director, Project AMP
Exploring Geometric Mean
Team Members: ______
File Name: ______
Lab Goal: To analyze the relationship between different sides and segments of right triangles.
Construction:
1. Construct AC . (Segment tool)
2. Draw a line through point C that is perpendicular to segment AC . (perpendicular line tool)
3. Label this line t. (label tool)
4. Place a point B on t. (Point tool)
5. Draw AB . (Segment tool)
Project AMP Dr. Antonio Quesada – Director, Project AMP
6. Draw a line from point C perpendicular to AB . (Perpendicular line tool)
7. Label the intersection point D. (Label tool)
8. DrawCD. (Segment tool)
suur 9. HideCD. (Hide/Show tool)
Tasks:
1. In the table below, record the indicated measures.
SEGMENT MEASURE
AC CB AD DB CD AB Table 1
2. Evaluate the following ratios:
a. AD b. CB c. AD CD AB AC
d. AC e. CD f. BD AB BD CB
3. Which of the above ratios are equivalent? Write the proportions below.
Project AMP Dr. Antonio Quesada – Director, Project AMP
4. Using the Tabulate tool create the table below. In the column headings, be sure to put the equivalent ratios next to each other. Then drag point C to four other positions and record the results in the table (press the tab key after each new position).
Table 2
5. Explain what you observed when point C was moved.
Definition: the geometric mean between two positive numbers a and b is the positive number X where a = X. X b
Now let’s answer some questions about the geometric mean.
Pick two ratios from Table 2 that are equal to each other.
6. Which segment is the geometric mean? Does this segment have a name? If so, what is the name of the segment? What are the other segments in your ratios? Do they have names? If so, what are their names?
Pick two more ratios that are equal to each other.
7. Which segment is the geometric mean? Does this segment have a name? If so, what is the name of the segment? What are the other segments in your ratios? Do they have names? If so, what are their names?
Project AMP Dr. Antonio Quesada – Director, Project AMP
Pick the last two ratios that are equal to each other.
8. Which segment is the geometric mean? Does this segment have a name? If so, what is the name of the segment? What are the other segments in your ratios? Do they have names? If so, what are their names?
Project AMP Dr. Antonio Quesada – Director, Project AMP
Exploring Geometric Mean: Sample Solution
Results:
1. Segment CD is the geometric mean of segments AD and BD. In other words, the altitude is the geometric mean of the two segments of the hypotenuse. 2. Segment AC is the geometric mean of segments AB and AD . In other words, the leg is the geometric mean of the hypotenuse and the segment of the hypotenuse adjacent to that leg. 3. Segment CB is the geometric mean of segments AB and BD. In other words, the leg is the geometric mean of the hypotenuse and the segment of the hypotenuse adjacent to that leg.