Topic: Visualizing Arithmetic and Geometric Means

Level:

The Geometer's SketchPad Proficiency:

Menus:

Students should be familiar with the Construct Menu

Notes to Teachers:

Students ought previously to be taught the algebraic definitions of the Arithmetic Mean = (a+b)/2 and the Geometric Mean = square root of (a*b).

Construction Guide:

1. Construct a line segment AC
   
   
2. Construct the midpoint M of segment AC
   
   
3. Construct a point B on segment AC
   
   
4. Using AB for the value a and BC for the value b, find the arithmetic mean and the geometric mean.
   
   
5. Construct a circle with M as the center and MC as the radius
   
   
6. Construct a perpendicular to AC at point B
   
   
7. Construct the point of intersection of this line with the circle
   
   
8. Hide the line and construct the segment BD.
   
   
9. Hide the extra point of intersection.
   
   
10. Repeat the above construction to obtain a perpendicular segment to AC at point M which intersects the circle at point N.
    
    

Explorations

1. Using the measure of AB as a and the measure of BC as b, find the arithmetic mean and the geometric mean.
   
   
2. Calculate the lengths of BD and MN.
   
   
3. Which length matches which mean?
   
   
4. Drag point B and observe what happens to the relationship between the length of BD and the length of MN. What conclusion do you find is true for the arithmetic mean and the geometric mean?

EXPLORE FURTHER:

• How can you prove those relationships?

Sister Myrona
Mount Saint Mary Academy
Watchung, New Jersey