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Geometry Day 55

Geometry Day 55. Geometric Mean. Today’s Agenda. Geometric Mean Geometric Mean in Right Triangles. Geometric Mean. A Geometric Mean is a kind of average. To find the Geometric Mean between two numbers, multiply them together and take the square root.

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Geometry Day 55

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  1. Geometry Day 55 Geometric Mean

  2. Today’s Agenda • Geometric Mean • Geometric Mean in Right Triangles

  3. Geometric Mean • A Geometric Mean is a kind of average. • To find the Geometric Mean between two numbers, multiply them together and take the square root. • Example: Find the Geometric Mean of 5 and 20.

  4. Practice • Turn to p. 97 of your workbook, and try problems 1-6. • Answers:

  5. Geometric Mean as a Proportion • In a proportion if the means are equal, then that value is the geometric mean of the extremes: • x represents the Geometric Mean between a and b. • Example:

  6. Geometric Mean • This concept can used in Geometry. • One particular use is when dealing with right triangles.

  7. Geometric Mean • We start with a Right Triangle

  8. Geometric Mean • Let’s draw its altitude.

  9. Geometric Mean • We’ve now formed 2 more triangles – 3 in all! • What do these 3 triangles have in common?

  10. Geometric Mean • Let’s consider the original diagram C A B D

  11. Geometric Mean • We’ll put the others up for reference A C A A B D C D B D B C C

  12. Geometric Mean • Let’s label the sides A C A a b h d e A B c D a c a d C b h D B D B C C h e b

  13. Geometric Mean • We can use similarity properties to set up proportions: A C A a b h d e A B c D a c a d C b h D B D B C C h e b

  14. Geometric Mean • To conclude, a, b, and h, can all be written as the Geometric Mean of two segments. a b h d e c

  15. Geometric Mean • Putting it in words: • The altitude drawn to the hypotenuse of a right triangle separates the hypotenuse into two segments. • The length of this altitude is the geometric mean between the lengths of these segments. • The length of a leg of this triangle is the geometric mean between the length of the hypotenuse and the segment of the hypotenuse adjacent to that leg.

  16. Example 5 20 x y z

  17. Homework 31 • Workbook, p. 98 • Word Problem Handout • If you have trouble remembering the proportions, try drawing and labeling the triangles separately. • You can confirm your answers with the Pythagorean Theorem.

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