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How to find the angles of a triangle in a semicircle?

  1. Jun 14, 2017 #1
    1. The problem statement, all variables and given/known data
    Screenshot_29_2014_mat_a.png

    2. Relevant equations
    d(y)/d(x) --> max area
    area of triangle = 1/2 . base . height




    3. The attempt at a solution
    for number (2)

    2_-_2014_mat_a.png
    x^2 + y^2 = r^2 --> circle equation
    base = 2R, height = y
    Area = 1/2 . 2R . y
    area = 1/2 . 4. √ (r^2 - x^2)

    area now is half of max = 2, so :
    2 = 1/2 . 4. √ (r^2 - x^2)
    2 = 2 √(4-x^2)
    1 = √ (4-x^2)
    1 = 4 - x^2
    x = √3
    now y = 1
    AP = 2 - √3
    I have to find w (angle of ACB)
    PC = 1
    how to find w ?
     
  2. jcsd
  3. Jun 14, 2017 #2

    Doc Al

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    Staff: Mentor

    Consider the triangle AOC.
     
  4. Jun 15, 2017 #3
    OC^2 = AC^2 + AO^2 -2AC.AO.cosw
    4 = 4 + (8-4v3)^2 -8(8-4v3)cosw
    Cos w = (8-4v3)/8
    Cos w = 1-1/2v3

    But the answer is in degrees, and not allowed using calculator.. And I dont think my answer is right.

    Note: v3 = root 3
     
  5. Jun 15, 2017 #4

    Doc Al

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    Consider the triangle AOC. You can find the angle at O by studying the right triangle CPO.
     
  6. Jun 15, 2017 #5
    I don't get it.. Can anyone help??
     
  7. Jun 15, 2017 #6

    ehild

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    upload_2017-6-16_5-18-8.png
    CPO is a right triangle. You know the hypotenuse, it is one radius, length 2. You calculated PC, it is 1. What is the blue angle then?
     
  8. Jun 15, 2017 #7
    The blue angle is 60degrees.
     
  9. Jun 15, 2017 #8

    ehild

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    No. Imagine half of an equilateral triangle, rotated :smile:
     
  10. Jun 15, 2017 #9
    Oh.. 30degrees ...
     
  11. Jun 15, 2017 #10
    So, i get it. One triangle = 180degrees. I get 30, left 150. CAO = ACO. So CAO = 75 degrees?
     
  12. Jun 15, 2017 #11
    But with theory, it should be sin w/ 2 = sin tetha / AC
    I get sin w = 1/AC
    While AC = 8-4root3
    How can i get w with this method?
     
  13. Jun 16, 2017 #12

    ehild

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    Yes, it is correct.
     
  14. Jun 16, 2017 #13

    ehild

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    What is theta, and what "theory" is that?
    Check with a calculator. Otherwise it is complicated.
     
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