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Warm-up: Simplify. 1) 2) 3). 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. B. D.
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Warm-up: Simplify • 1) • 2) • 3)
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.
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
So ABC~ DAC B D B D A A C D C C A
B D A C Similarly, the small triangle must be flipped to match up the angles. B D B D A C A
So ABC~ DBA B D B A C D A B D C A
B A D C A C And DAC~ DBA By Transitive Property B D A
ABC~ DAC ABC~ DBA DAC~ DBA 1)2) Ex: Use the similarity statements to complete AD AC
3)4) AC AC BC BAC or ADC 5) BDA~
Arithmetic Mean of two numbers = (average) Geometric Mean of two numbers – is the positive number x such that
Answer: Ex: Find the Geometric mean between each set of numbers: 1) 2) Answer:
Answer: x = 3) 4) Answer: x =
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…
EX: Find the arithmetic mean of the numbers 2, 3 and 4 Ex: Now find the geometric mean of 2, 3 and 4
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
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
H Ex: Find HF 9 GF = 4 G 6 HF = 13 E F
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
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
H Ex: Find HE, EF and EG in simplest form 8 EG = EF = G 5 HE = E F