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Understanding Geometric Mean in Mathematics

Explore the concept of geometric mean between two numbers and its relevance in geometry. Discover how altitude drawn in a right triangle relates to geometric means. Practice finding geometric means with provided examples. Learn how to identify the "W" pattern and apply proportions to solve for geometric means. Dive into Theorem 7.1 and Theorem 7.2 to deepen your understanding.

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Understanding Geometric Mean in Mathematics

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  1. Lesson 7-1 Geometric Mean Lesson 7-1: Geometric Mean

  2. The geometric mean between two numbers a and b is the positive number x where . Therefore x = . Try It: Find the geometric mean of . . . Answer = 20 1. 10 and 40 2. 1 and 36 Answer = 6 3. 10 and 20 Answer = 14.14 4. 5 and 6 Answer = 5.48 Answer = 9.94 5. 8.1 and 12.2 Lesson 7-1: Geometric Mean

  3. A B C How does this relate to geometry? Lesson 7-1: Geometric Mean

  4. If the altitude is drawn from the vertex of the 90° angle of a Right ▲ to its hypotenuse, then the two new ▲s formed are similar to the original ▲ andto each other.Example: ▲ABC ~ ▲DBA ~ ▲DAC Theorem 7.1 Lesson 7-1: Geometric Mean

  5. The " W " Pattern Lesson 7-1: Geometric Mean

  6. The Geometric Means Recall the three geometric means that you discovered from your Sketchpad activity. BUT FIRST . . . Lesson 7-1: Geometric Mean

  7. Re-label the Sides (as lengths) Lesson 7-1: Geometric Mean

  8. Geometric Mean #1 What is the proportion that uses f? f is the geometric mean of d and e. Lesson 7-1: Geometric Mean

  9. Geometric Mean #2 What is the proportion that uses b? b is the geometric mean of e and c. Lesson 7-1: Geometric Mean

  10. Geometric Mean #3 What is the proportion that uses a? a is the geometric mean of d and c. Lesson 7-1: Geometric Mean

  11. Put them all together Lesson 7-1: Geometric Mean

  12. The “W” Pattern W Lesson 7-1: Geometric Mean

  13. The measure of an altitude drawn from the vertex of the 90° angle of a Right ▲ to its hypotenuse is the geometric mean between the measures (lengths) of the two segments of the hypotenuse.Example: AD is the geometric mean of BD and DC. Theorem 7.2 Lesson 7-1: Geometric Mean

  14. Try it ! Given: d = 4 and e = 10 Find: a = ___ b = ___ c = ___ f = ___ Lesson 7-1: Geometric Mean

  15. Solution: Proportions Answers Lesson 7-1: Geometric Mean

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