Geometry 8.1 Right Triangles

A radical is in simplest form when: No perfect square factor other than 1 is under the radical sign. No fraction is under the radical sign. No radical is in a denominator. #2 and #4 Together…You got the rest!! Simplify: Answers:

Geometric Mean If a, b and x are positive numbers and mean mean then x is called the geometric mean between a and b. Multiplying means/extremes we find that: Taking the square root of each side: A positive number

Geometric Mean Basically, to find the geometric mean of 2 numbers, multiply them and take the square root. OR (add to notes: take the square root of each, then multiply try to find the geometric mean between 4 and 9 and the geometric mean between 36 and 50) Note that the geometric mean always falls between the 2 numbers. Ex: Find the geometric mean between 5 and 11. Use the formula

Geometric Mean Ex: Find the geometric mean between the two numbers. a. 5 and 20 Avoid multiplying large numbers together. Break numbers into perfect square factors to simplify. b. 24 and 32

Directions: Find the geometric mean between the two numbers. 8. 64 and 49 9. 1 and 3 10. 100 and 6 11. 20 and 24

Geometric Mean Find the geometric mean between the two numbers. 5 and 20 64 and 49 1 and 3 100 and 6 20 and 24

∆ Review Hypotenuse the side opposite the right angle in a right triangle Altitude the perpendicular segment from a vertex to the line containing the opposite side

Theorem 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. ~ a a ~ b b

Corollaries When the altitude is drawn to the hypotenuse of a right triangle: Y the length of the altitude is the geometric mean between the segments of the hypotenuse. X A Z Each leg is the geometric mean between the hypotenuse and the segment of the hypotenuse that is adjacent to that leg.

Corollary 1 = piece of hypotenuse altitude altitude other piece of hypotenuse = Y X A Z

Corollary 2 hypotenuse leg leg piece of hyp. adj. to leg = Y X A Z

Corollary 2 hypotenuse leg leg piece of hyp. adj. to leg = Y X A Z

Directions: Exercises 24-31 refer to the diagram at right. 12. If CN = 8 and NB = 16, find AN. 13. If AN = 4 and CN = 12, find NB. 14. If AN = 4 and CN = 8, find AB. 15. If AB = 18 and CB = 12, find NB. 16. If AC = 6 and AN = 4, find NB. N A C B

Homework pg. 288 #16-30, 31-39 odd

Exercises C 8 A x N 16 B 12. If CN = 8 and NB = 16, find AN. Let x = AN

Exercises C x A N B 8 12 14. If AN = 8 and NB = 12, find CN. Let x = CN

Exercises C 12 x A N B 18 17. If AB = 18 and CB = 12, find NB. Let x = NB

Answers Exercises 13 - 19 36 20 2√15 25 5