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Special Right Triangles

Special Right Triangles. Unit 4. Simplifying Radicals. √ radical Radicand – number inside the radical http://www.youtube.com/watch?v=HU5IawUD2o8 You can click on other videos for more explainations. √. Examples. √6 ∙ √8 √2∙2∙2∙2∙3 4√3 2) √90 √2∙3∙3∙5

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Special Right Triangles

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  1. Special Right Triangles Unit 4

  2. Simplifying Radicals • √ radical • Radicand – number inside the radical • http://www.youtube.com/watch?v=HU5IawUD2o8 • You can click on other videos for more explainations. √

  3. Examples • √6 ∙ √8 √2∙2∙2∙2∙3 4√3 2) √90 √2∙3∙3∙5 3√10 3) √243 √3 √3∙3∙3∙3∙3 √3 9 √3 √3 9

  4. Division – multiply numerator and denominator by the radical in the denominator 4) √25 √3 5 ∙√3 √3 ∙√3 5 ∙√3 3 • 8 = √14 √ 28 7 6) √5 ∙ √35 √14 √5∙5 ∙7 √2∙7 5√7 √2∙7 √2∙7 √2∙7 35 √2 = 5 √2 14 2

  5. What have you learned today? What is still confusing?

  6. Pythagorean Theorem and It’s Converse Objective: to use the Pythagorean Theorem and it’s converse. c2 = a2 + b2

  7. Key Concept • Pythagorean Theorem c2 =a2 + b2 c - hypotenuse a – altitude leg b – base leg

  8. Key Concepts B a c C b A Acute c2 < a2 + b2 Right c2 = a2 + b2 Obtuse c2 > a2 + b2 B a c C b A B a c C b A

  9. Ex 1 Find the value of x. Leave in simplest radical form. Answer: 2 √11 x 12 10

  10. Ex 2: Baseball A baseball diamond is a square with 90 ft sides. Home plate and second base are at opposite vertices of the square. About far is home plate from second base to the nearest foot? About 127 ft

  11. Ex 3:Classify the triangle as acute, right or obtuse. • 15, 20, 25 right b) 10, 15, 20 Obtuse

  12. Pythagorean Triplet Whole numbers that satisfy c2 = a2 + b2. Example: 3, 4, 5 Can you find another set?

  13. What have you learned today? What is still confusing?

  14. Special Right Triangles Objective: To use the properties of 45⁰ – 45⁰ – 90⁰ and 30⁰ – 60⁰ - 90⁰ triangles.

  15. Isosceles Right TriangleKey Concept 45⁰-45⁰ -90⁰ X – X – X2 102 22 2 10 2 10

  16. Example 1 Find x. Simplify. 45 – 45 – 90 x – x - x 2 8 – 8 – 82 82

  17. Try 1:FIND THE MISSING LENGTHS. SIMPLEST RADICAL FORM. 45⁰ 45⁰ 90⁰ X X X2 1212 X2 = 1212 2 2 X = 121 MISSING SIDE LENGTHS ARE 121. 1212 45⁰

  18. Example 2 Find x. Simplify. 45 – 45 – 90 y – y - y 2 – – 28 28 = y 2 set up equation 2 2 divide by root 2 y = 28 2 2 2 = 28 2 2 = 142

  19. Try2: Find the length of the hypotenuse of a 45⁰ – 45⁰ – 90⁰ triangle with legs of length 5√6 . 45⁰ – 45⁰ – 90⁰ x - x - x√2 X = 5√6 x√2 = 5√6√2 substitute into the formula = 10 √3

  20. Try 3 Find the length of a leg of a 45⁰ – 45⁰ – 90⁰ triangle with hypotenuse of length 22. 45⁰ – 45⁰ – 90⁰ x - x - x√2 x√2 = 22 solve for x X = 22 = 22√2 = 11√2 √2 2

  21. Try 4: The distance from one corner to the opposite corner of a square field is 96ft. To the nearest foot, how long is each side of the field? 45⁰ – 45⁰ – 90⁰ x - x - x√2 x√2 = 96 solve for x X = 96 = 96√2 = 48√2 √2 2

  22. What is the relationship of the legs and hypotenuse of an isosceles right triangle? 45⁰ – 45⁰ – 90⁰ x - x - x√2

  23. Equilateral Triangle Key Concept 30⁰ - 60⁰ - 90⁰ X – X3 – 2X 30⁰ 4 2 3 60⁰ 2

  24. Example 3 Solve for missing parts of each triangle: x = 10 y = 5√3 x y 5

  25. Example 4 The longer leg of a 30⁰ – 60⁰ - 90⁰ triangle has length of 18. Find the lengths of the shorter leg and the hypotenuse. 30⁰ – 60⁰ - 90⁰ x - x√3 - 2x x√3 = 18 solve for x X = 18 = 18√3 = 6√3 – short leg √3 3 12√3 - hypotenuse

  26. Try 5 30 - 60 - 90 X - X3 - 2X 9 2X = 9 X = 9/2 a = 9/2 b = 93 2 30 - 60 - 90 X - X3 - 2X 25 25 = x 3 3 3 X = 25 3 3 d = 25 3 3 c = 50 3 3 3 3

  27. What is the relationship of the 30-60-90 right triangle? 30⁰ – 60⁰ - 90⁰ x - x√3 - 2x

  28. Exit Ticket X = 5 Y = 5 2 Z = 53 3 W = 103 3

  29. What have you learned today? What is still confusing? Click on the link.

  30. Similarities in Right Triangles Objective: To find and use relationships in similar right triangles

  31. Geometric mean with similar right triangles

  32. Type 1 Relationship: altitude side2 side 1 altitude = altitude Side 1 Side 2

  33. Example 1

  34. Try 1:

  35. Type 2 relationship: Hypotenuse Leg(hyp) Leg ______ side = Big Small Leg side Hypotenuse

  36. Example 2:

  37. Example 3:

  38. Try: 2

  39. Try 3:

  40. Try 4:

  41. Exit Ticket Find x, y, and z. X = 6 9 x 36 = 9x 4 = x 9 = z z 9+x Z ²= 9(13) Z = 3√13 y = x 9+x y Y ² = 4(13) Y = 2√13

  42. 8-3 and 8-4Right Triangle Trigonometry Objective • To use sine, cosine and tangent ratios to determine side and angle measures in triangles

  43. Key Concept Sine (sin) Cosine(cos) Tangent(tan) SOH CAH TOA The Old Aunt Sat On Her Coat and Hat

  44. Example 1 • Find sin E, cos E, and tan E. sin E = 8/10 = 4/5 cos E = 6/10 = 3/5 tan E =8/6 = 4/3 Try: Find sin F, cos F, and tan F. sin F = 6/10 = 3/5 cos F = 8/10 = 4/5 tan F =6/8 = 3/4 E 6 10 G 8 F

  45. Example 2 – round to the nearest tenths. OPP/ADJ Tan (37⁰) = x/3 3(tan (37⁰)) = x X 2.3 Opp adj

  46. Try 1 OPP/HYP Sin (67⁰) = 10/x 10/sin (67⁰) = x X  10.9 Opp hyp

  47. TRY 2 • Adj/hyp • Cos (40⁰) = 6/x • 6/cos(40⁰) = x • X 4.6 adj hyp

  48. Ex 3:When solving for angles use inverse.Round to the nearest degree Adj/hyp Cos(x) = 8/18 Cos-1(8/18)  64⁰

  49. TRY Opp/adj Tan (x) = 11/8 tan-1(11/8)  54⁰

  50. What have you learned today? What is still confusing?

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