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Warm-up: Simplify

Understand the relations between altitudes, hypotenuses, and segments in right triangles using geometric means and similarity theorems. Explore how to apply these concepts in finding heights, means of numbers, and geometric shapes.

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Warm-up: Simplify

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  1. Warm-up: Simplify • 1) • 2) • 3)

  2. B D ABC~ DAC ABC~ DBA DAC~ DBA A C 9.1 – Exploring Right Triangles Theorem 9.1 – If an 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.

  3. B D A C Start with large ABC and altitude AD To match the angles, the medium triangle must be flipped over. B D D A C C A

  4. So ABC~ DAC B D B D A A C D C C A

  5. B D A C Similarly, the small triangle must be flipped to match up the angles. B D B D A C A

  6. So ABC~ DBA B D B A C D A B D C A

  7. B A D C A C And DAC~ DBA By Transitive Property B D A

  8. ABC~ DAC ABC~ DBA DAC~ DBA 1)2) Ex: Use the similarity statements to complete AD AC

  9. 3)4) AC AC BC BAC or ADC 5) BDA~

  10. Arithmetic Mean of two numbers = (average) Geometric Mean of two numbers – is the positive number x such that

  11. Answer: Ex: Find the Geometric mean between each set of numbers: 1) 2) Answer:

  12. Answer: x = 3) 4) Answer: x =

  13. So, another way to think of geometric mean: The geometric mean of a and b = The geometric mean of a, b and c = And so on…

  14. EX: Find the arithmetic mean of the numbers 2, 3 and 4 Ex: Now find the geometric mean of 2, 3 and 4

  15. Theorem 9.2 – In a rt. Triangle, the length of the altitude to the hypotenuse is the geometric mean of the length of the two segments of the hypotenuse. C B D A

  16. C B D A Theorem 9.2 – In a rt. Triangle, the length of the altitude to the hypotenuse is the geometric mean of the length of the two segments of the hypotenuse. BD BD

  17. H Ex: Find HF 9 GF = 4 G 6 HF = 13 E F

  18. Ex: To find the height of Ms. Van Horn’s room, Mike holds a book so that the corner of the ceiling and floor are in line with the edges of the book. If Mike’s eye is 5 feet from the floor, and he is standing 14 feet away from the wall, how high is the wall? x X = 39.2 14 Wall is 44.2 ft. high 5

  19. C B D A Theorem 9.3 – Each leg of the right triangle is the geometric mean of the whole hypotenuse and the segment of the hypotenuse that is adjacent to the leg. CA DA CA DC

  20. H Ex: Find HE, EF and EG in simplest form 8 EG = EF = G 5 HE = E F

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