Geometry 8 1 Right Triangles A radical is

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Geometry 8. 1 Right Triangles

Geometry 8. 1 Right Triangles

A radical is in simplest form when: 1. No perfect square factor other than

A radical is in simplest form when: 1. No perfect square factor other than 1 is under the radical sign. 2. No fraction is under the radical sign. 3. No radical is in a denominator. Simplify: Answers: #2 and #4 Together…You got the rest!!

Geometric Mean If a, b and x are positive numbers and mean then x

Geometric Mean If a, b and x are positive numbers and 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

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

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.

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. 7. 5 and 20

Geometric Mean Find the geometric mean between the two numbers. 7. 5 and 20 8. 64 and 49 9. 1 and 3 10. 100 and 6 11. 20 and 24

∆ Review Hypotenuse the side opposite the right angle in a right triangle •

∆ 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

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 b a ~ b

Corollaries When the altitude is drawn to the hypotenuse of a right triangle: Y

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 Each leg is the geometric mean between the hypotenuse and the segment of the hypotenuse that is adjacent to that leg. A Z

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

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

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

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

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

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

Directions: Exercises 24 -31 refer to the diagram at right. 12. If CN =

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. C A N B

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

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

Exercises C 8 A x 12. If CN = 8 and NB = 16,

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

Exercises C x A 8 14. If AN = 8 and NB = 12,

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

Exercises C 12 N x 18 17. If AB = 18 and CB =

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

Answers Exercises 13 - 19 13. 15. 16. 18. 19. 36 20 2√ 15

Answers Exercises 13 - 19 13. 15. 16. 18. 19. 36 20 2√ 15 25 5

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