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Right Triangles and Trigonometry Chapter 8

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8.1 Geometric Mean Geometric mean: Ex: Find the geometric mean between 5 and 45 Ex: Find the geometric mean between 8 and 10

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If an altitude is drawn from the right angle of a right triangle. The two new triangles and the original triangle are all similar. A D C B

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The altitude from a right angle of a right triangle is the geometric mean of the two hypotenuse segments A D C B Ex:

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The leg of the triangle is the geometric mean of the hypotenuse and the segment of the hypotenuse adjacent A D C B Ex:

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8.2 Pythagorean Theorem and its Converse How, when and why do you use the Pythagorean Theorem and its converse? How: the square of the two legs added together equals the hypotenuse squared When: given a right triangle and the length of any two sides Why: to find the length of one side of a right triangle

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Pythagorean Theorem: When c is unknown: When a or b is unknown: 5 3 x 7 x 14 c a b

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Converse: the sum of the squares of 2 sides of a triangle equal the square of the longest side 8, 15, 16 Pythagorean Triple: 3 lengths that always make a right triangle 3, 4, 5 5, 12, 13 7, 24, 25 9, 40, 41 Not =, so not a right triangle

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8.3 Special Right Triangles 30-60-90 Short leg is across from the 30 degree angle Long leg is across from the 60 degree angle Ex: 30 14 y x

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45-45-90 The legs are congruent Ex: x x 6 x 8

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8.6 Law of Sines Use two of the ratios to make a proportion and solve To solve a triangle: means to solve for all missing angles and sides c b a AC B

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Solve the triangle 33 47 14 B A C

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8.7 Law of Cosines c b a AC B

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Solve the Triangle A B C 60 10 8

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