The Geometric Mean

Means and Extremes

• From the Cross-Product Property, you may state this as “The product of the means is equal to the product of the means.”

The Geometric Mean

• For any two positive numbers a and b, the geometric mean of a and b is the positive number x such that .

• Using cross multiplication, you can find

Example of Geometric Mean

• What is the geometric mean of 12 and 3?• or • By cross multiplying: • Therefore Check:• or

Right Triangles & the Geometric Mean

• As you have discovered, in a right triangle, the altitude to the hypotenuse yields three similar triangles.

• The length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the segments of the hypotenuse.

ExampleYou should see three triangles.From the work you have done, you should know all 3 are similar.As for corresponding parts, you may just see “shorter leg”, “longerleg” and “hypotenuse”.

Drawing them separatelymay help you see the corresponding parts.You could also just list the parts as proportionsas you did in the activity.

Example 2

To solve

Homework clue for p.443, #41-48

Not Drawn to Scale; Only as a reference

𝒽2

𝒽1

𝒶ℓ2

ℓ1

𝒽2+𝒽1=𝓱

Homework• Prepare for quiz next class– Review Last Quiz (in mailboxes)– Review homework from last class and below– Prepare questions for next class (pre-quiz Q&A)

• Pp. 442– 9 – 20 – 23a– 25– 26 – 33 (review Ex. 1 pg 440)– 34 – 36– 41 – 48