1 / 21

7.1 Geometric Mean

7.1 Geometric Mean. Objectives. Find the geometric mean between two numbers Solve problems involving relationships between parts of right triangles and the altitude to its hypotenuse. Geometric Mean. The geometric mean between two numbers is the positive square root of their products.

sadie
Download Presentation

7.1 Geometric Mean

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 7.1 Geometric Mean

  2. Objectives • Find the geometric mean between two numbers • Solve problems involving relationships between parts of right triangles and the altitude to its hypotenuse

  3. Geometric Mean • The geometric mean between two numbers is the positive square root of their products. • In other words, given two positive numbers such as a and b, the geometric mean is the positive numberxsuch that a : x= x : bWe can also write this as fractions, a / x = x / bor as cross products, x 2= ab.

  4. Example 1a: Find the geometric mean between 2 and 50. Let x represent the geometric mean. Definition of geometric mean Cross products Take the positive square root of each side. Simplify. Answer: The geometric mean is 10.

  5. Example 1b: Find the geometric mean between 25 and 7. Let x represent the geometric mean. Definition of geometric mean Cross products Take the positive square root of each side. Simplify. Use a calculator. Answer: The geometric mean is about 13.2.

  6. Your Turn: a. Find the geometric mean between 3 and 12. b. Find the geometric mean between 4 and 20. Answer: 6 Answer: 8.9

  7. Theorem 7.1 • If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other. ∆CBD ~ ∆ABC ∆ACD ~ ∆ABC ∆CBD ~ ∆ACD

  8. Geometric Mean Theorems • Theorem 7.2:In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments. The length of the altitude is the geometric mean of the lengths of the two segments. • Theorem 7.3:In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments. The length of each leg of the right triangle is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse that is adjacent to the leg. BD CD = CD AD AB CB = CB DB AB AC = AC AD

  9. What does all this mean? 6 x 5 + 2 y = = x 3 y 2 7 y 18 = x2 = y 2 √18 = x 14 = y2 √9 ∙ √2 = x √14 = y 3 √2 = x

  10. Example 2:

  11. Example 2: Cross products Take the positive square root of each side. Use a calculator. Answer: CD is about 12.7.

  12. Your Turn: Answer: about 8.5

  13. Example 3: KITES Ms. Turner is constructing a kite for her son. She has to arrange perpendicularly two support rods, the shorter of which is 27 inches long. If she has to place the short rod 7.25 inches from one end of the long rod in order to form two right triangles with the kite fabric, what is the length of the long rod?

  14. Let be the altitude drawn from the right angle of Example 3: Draw a diagram of one of the right triangles formed.

  15. Example 3: Cross products Divide each side by 7.25. Answer: The length of the long rod is 7.25 + 25.2, or about 32.4 inches long.

  16. AIRPLANES A jetliner has a wingspan, BD, of 211 feet. The segment drawn from the front of the plane to the tail, intersects at point E. If AE is 163 feet, what is the length of the aircraft? Your Turn: Answer: about 231.3 ft

  17. Find c and d in Example 4:

  18. is the altitude of right triangle JKL. Use Theorem 7.2 to write a proportion. Example 4: Cross products Divide each side by 5.

  19. is the leg of right triangle JKL. Use the Theorem 7.3 to write a proportion. Answer: Example 4: Cross products Take the square root. Simplify. Use a calculator.

  20. f Answer: Your Turn: Find e and f.

  21. Assignment • Pre-AP Geometry Pg. 346 #13 – 38 & #44 • Geometry Pg. 346 #13 – 32, #35 - 38

More Related