Warm-up: Simplify 1) 1) 2) 2) 3) 3)

A C B D 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. ABC~ DAC ABC~ DBA DAC~ DBA

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

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

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

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

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

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

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

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

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

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 2, 3 and 4 Ex: Now find the geometric mean of 2, 3 and 4 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 BD

Ex: Find HF HF = 13 GF = 4 H G FE 9 6

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? 5 14 x X = 39.2 Wall is 44.2 ft. high

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. C B D A CA DA CA DC

Ex: Find HE, EF and EG in simplest form EF =EG = H G FE 8 5 HE =