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CHAPTER EIGHT

CHAPTER EIGHT. Alec Rodriguez Jack Wells Chris “the Bottman ” Bott. 8.1 Similarity in Right Triangles. Theorem 8-1 Right Triangle Similarity 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.

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CHAPTER EIGHT

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  1. CHAPTER EIGHT Alec Rodriguez Jack Wells Chris “the Bottman” Bott

  2. 8.1 Similarity in Right Triangles • Theorem 8-1 Right Triangle Similarity • 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. A D B C

  3. Geometric Mean • The mean between two numbers in a geometric sequence. • 2,4,8,16,32 • a/x =x/b • Ex. 2/x = x/32 • Answer: 8

  4. Corollary 1 • When the altitude is drawn to the hypotenuse of a right triangle, the length of the altitude is the geometric mean between the segments of the hypotenuse. A AD/CD = CD/BD D B C

  5. Corollary 2 • When the altitude is drawn to the hypotenuse of a right triangle, each leg is the geometric mean between the hypotenuse and the segment of the hypotenuse that is adjacent to that leg. A AB/AC= AC/AD D B C AB/BC = BC/BD

  6. Challenge Find AB, AC, CD, CB Triangle ABC is a right triangle. A 9 D 16 X Y B C Z

  7. The Pythagorean Theorem • In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs. • A2 + B2 = C2 • Pythagoras.

  8. Challenge 1. Find X. 2. Find C. 90° 2√2 45° X C 6 90° 8

  9. 8-3 Converse of the Pythagorean Theorem B A C C2 = A2 + B2 Right Triangle C2 < A2 + B2 Acute Triangle C2 > A2 + B2 Obtuse Triangle

  10. 8-3 Converse of the Pythagorean Theorem Example: Is the triangle acute, obtuse, or right? 132 + 152 ____ 292 169 + 225 ____ 841 394 < 841 The triangle is acute. 122 + 182____ 192 144 + 324 ____ 361 468 > 361 The triangle is obtuse. 12 18 13 15 19 29

  11. Special Right Triangles 1. 45 – 45 – 90 General Rule 45 a 2 a 45 a

  12. More Special Right Triangles 2. 30 – 60 – 90 General Rule 60 2a a 30 a 3

  13. Even More Special Right Triangles Challenge Find X. 7 45 X 30 Y Find Y. 4 3

  14. Sine • Formula : sinѲ=Opposite Hypotenuse • Solve for x: • Sin20=4/x • Multiply each side by x • X(sin20)=4 • Divide each side by sin20 • X=11.695 x Hypotenuse Opposite 4 20⁰ Ѳ Adjacent

  15. Cosine • Formula : cosѲ=Adjacent Hypotenuse • Solve for x: • Cos67=x/120 • Multiply both sides by 120 • 120(cos67)=x • Multiply 120 and cos67 • 46.88=x Hypotenuse Opposite 120 x 67⁰ Ѳ Adjacent

  16. Tangent • Formula : tanѲ=Opposite Adjacent • Solve for x • Tan42=x/5 • Multiply each side by 5 • 5(tan42)=x • Multiply 5 and tan42 • 4.5=x x Hypotenuse Opposite 5 42⁰ Ѳ Adjacent

  17. SOH-CAH-TOA • An easy way to remember all of these formulas is by using SOH CAH TOA • SOH - (sine) opposite over hypotenuse • CAH - (cosine) adjacent over hypotenuse • TOA - (tangent) opposite over adjacent

  18. Applications of Right Triangle Trigonometry Angle of depression • How to solve: • Tan2⁰ = 25/x • x = 25/tan2⁰ • x = 716.3 horizontal 2⁰ Line of sight 25 2⁰ x horizontal Angle of elevation

  19. Exercises When the sun’s angle of elevation is 57⁰, a building casts a shadow 21m long. How high is the building? • Solve for x and y: 35 x 37⁰ y 57⁰ 21m

  20. Last Exercise • An observer is located 3km from a rocket launch site sees a rocket at an angle of elevation of 38⁰. How high is the rocket at that moment?

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