Geometric Mean

by

Mike Rosonet

Goal: Students will come to understand what the geometric mean is numerically and geometrically.

Important Information: the definition of geometric mean:

Geometric mean - the positive square root of the product of 2 numbers

Example 1: Determine the geometric mean of the following pairs of numbers. Let x be the geometric mean.

a. 2 and 50

b. 25 and 7

Students will need to recall the similarity of triangles, specifically, the proportions made between them.

Given right triangle ABC with perpendicular from B to side AC meeting at point D,

the following statement and proportion are each true:

Therefore, through multiplication and simplification, the following conclusion can be made:

Therefore, by the definition of geometric mean, the segment BD is the geometric mean of the two triangles. This means that, given that two triangles are similar, a similar proportion can be created to solve for any given side or for the geometric mean of the triangles.

Example 2: Find the length of segment YZ.

Thus, the length of segment YZ, which is the geometric mean of the two triangles in the figure, is approximately 12.7.

Students will need to know that proportions can also be made of ratios in the following formats:

and

Have students come up to the SMART board (provided that the classroom has one) and label two lengths of pre-constructed similar right triangles using Geometer's Sketchpad. If necessary, use the GSP file provided here. Be certain that the students label two lengths that are found in one of the above three ratios.

Homework: (If all students have home access to the Internet, place this problem on class website. Have them print off the problem at home and work on it there. Otherwise, print this problem out as a worksheet and have them return it the next class period.)

Instructions: Find x and y.

Hint: Try using the following ratio. See what happens...

Solution: this would not be posted on a class website so that students could see it so clearly

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