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anonymous

  • one year ago

Please Help!!!!!

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  1. anonymous
    • one year ago
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    O is the center of the semicircle. If angle BCO=30 degrees and BC = \[6\sqrt{3}\] what is the area of triangle ABO?

  2. anonymous
    • one year ago
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    |dw:1435335512997:dw|

  3. misssunshinexxoxo
    • one year ago
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    Do you know how to find the area of a triangle?

  4. anonymous
    • one year ago
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    yes

  5. anonymous
    • one year ago
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    bh/2

  6. anonymous
    • one year ago
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    help someone...

  7. anonymous
    • one year ago
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    @misty1212

  8. mathstudent55
    • one year ago
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    |dw:1435338018836:dw|

  9. mathstudent55
    • one year ago
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    |dw:1435338107837:dw|

  10. mathstudent55
    • one year ago
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    Do you understand so far?

  11. anonymous
    • one year ago
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    wait how do we know it's 60?

  12. anonymous
    • one year ago
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    what about the triangle on the right side what do those angles equal to?

  13. mathstudent55
    • one year ago
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    I'll show you each angle I found, one at a time, and will explain.

  14. anonymous
    • one year ago
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    okay

  15. mathstudent55
    • one year ago
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    Start here. We have the side that is \(6 \sqrt 3\), the angle of 60 deg, and all three radii are congruent. |dw:1435338257728:dw|

  16. anonymous
    • one year ago
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    okay

  17. mathstudent55
    • one year ago
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    Now look at angle ABC. It is an inscribed angle. An inscribed angle is half the measure of the intercepted arc. Arc AC (on the bottom side) is 180 deg because it is a semicircle since we have the diameter AC.

  18. mathstudent55
    • one year ago
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    That makes angle ABC 90 degrees, and triangle ABC a right triangle.

  19. mathstudent55
    • one year ago
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    |dw:1435338496200:dw|

  20. anonymous
    • one year ago
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    right i remember all inscribed angles are 90 degrees

  21. mathstudent55
    • one year ago
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    Now look at triangle ABC. One angle is 90 deg. One angle is 30 deg. That means the left angle, angle A is 60 deg.

  22. mathstudent55
    • one year ago
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    |dw:1435338551348:dw|

  23. anonymous
    • one year ago
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    30 60 90 yea

  24. mathstudent55
    • one year ago
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    Correct.

  25. mathstudent55
    • one year ago
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    Now look at the congruent sides of triangle AOB.

  26. mathstudent55
    • one year ago
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    |dw:1435338598897:dw| Triangle AOB is isosceles. Opp angles to the congruent sides are congruent.

  27. mathstudent55
    • one year ago
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    |dw:1435338640262:dw|

  28. mathstudent55
    • one year ago
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    Once you have two angles of triangle AOB are 60 deg each, the third angle must also be 60 deg, and it's an equilateral triangle.

  29. anonymous
    • one year ago
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    sorry i lost you

  30. mathstudent55
    • one year ago
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    |dw:1435338690645:dw|

  31. anonymous
    • one year ago
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    Triangle AOB is isosceles. Opp angles to the congruent sides are congruent.

  32. anonymous
    • one year ago
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    what do u mean by that

  33. mathstudent55
    • one year ago
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    Did you understand that angle A of triangle AOB is 60 deg?

  34. anonymous
    • one year ago
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    yes

  35. mathstudent55
    • one year ago
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    There is a theorem that states: If a triangle has two congruent sides, then the angles opposite those sides are congruent.

  36. mathstudent55
    • one year ago
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    That means, in an isosceles triangle, the base angles are congruent.

  37. anonymous
    • one year ago
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    hmmmm.... okay

  38. mathstudent55
    • one year ago
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    |dw:1435338811305:dw|

  39. anonymous
    • one year ago
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    yes angles B and C are equal

  40. anonymous
    • one year ago
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    but the A is different....

  41. anonymous
    • one year ago
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    so how does it relate

  42. mathstudent55
    • one year ago
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    Ok. Now back to our problem. Notice that triangle AOB has two radii as sides, AO and OB. All radii in a circle are congruent, so sides AO and OB are congruent sides in a triangle.

  43. mathstudent55
    • one year ago
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    That means the base angles, BAO and ABO are congruent angles.

  44. mathstudent55
    • one year ago
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    |dw:1435339099484:dw|

  45. mathstudent55
    • one year ago
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    We now know that angle ABO also measures 60 deg. |dw:1435339121331:dw|

  46. mathstudent55
    • one year ago
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    Once two angles of a triangle are known, you can find the third angle. 60 + 60 + m<AOB = 180 m<AOB = 60

  47. mathstudent55
    • one year ago
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    Now we know that triangle AOB is also equilateral in addition to being isosceles.

  48. anonymous
    • one year ago
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    Cant it be this? |dw:1435339216515:dw|

  49. anonymous
    • one year ago
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    then it wouldn't be 60 60 60

  50. mathstudent55
    • one year ago
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    According to the given info, it must be equilateral.

  51. anonymous
    • one year ago
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    okay thank you

  52. anonymous
    • one year ago
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    im gonna have to look over this a few times im still having a hard time understanding why its all 60

  53. mathstudent55
    • one year ago
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    This is not the end of the problem. We still need to find the area of triangle ABO.

  54. mathstudent55
    • one year ago
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    Triangle ABC is a 30-60-90 triangle. AB is the short leg, and BC is the long leg. From the ratio of the lengths of the sides of a 30-60-90 triangle, we have: \(1~:~ \sqrt 3~:~ 2\) for the ratio of short leg : long leg : hypotenuse

  55. mathstudent55
    • one year ago
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    We see that the long leg is \(\sqrt 3\) times the length of the short leg. That means the short leg is \(\sqrt 3\) times shorter than the long leg.

  56. mathstudent55
    • one year ago
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    For triangle ABC, the short leg is AB, and the long leg is BC. Since AB is \(\sqrt 3\) times shorter than BC, AB is 6 units long.

  57. mathstudent55
    • one year ago
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    |dw:1435339904347:dw|

  58. mathstudent55
    • one year ago
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    Now the problem is simply to find the area of an equilateral triangle whose side has a length of 6.

  59. anonymous
    • one year ago
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    so bh/2 right?

  60. mathstudent55
    • one year ago
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    Yes, but we need to find the height. Let's just look at the triangle now.

  61. anonymous
    • one year ago
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    6 would be the opposite of 30 degrees

  62. anonymous
    • one year ago
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    and height is given as \[6\sqrt{3}\]

  63. mathstudent55
    • one year ago
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    |dw:1435339974085:dw|

  64. anonymous
    • one year ago
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    ohhhhh yeaaaa

  65. mathstudent55
    • one year ago
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    |dw:1435340010942:dw|

  66. mathstudent55
    • one year ago
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    Inside triangle AOB, we have two 30-60-90 triangles. |dw:1435340060713:dw|

  67. mathstudent55
    • one year ago
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    XO is 3, since 3 is half of 6. Now we need h, the height of triangle AOB. h is also the long leg of the 30-60-90 triangle BXO.

  68. anonymous
    • one year ago
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    so it would be \[9\sqrt{3}\]

  69. mathstudent55
    • one year ago
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    Remember, the long leg is \(\sqrt 3\) times longer than the short leg. |dw:1435340159438:dw|

  70. mathstudent55
    • one year ago
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    |dw:1435340236323:dw|

  71. anonymous
    • one year ago
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    i understand everything except how it's a 60-60-60- triangle

  72. mathstudent55
    • one year ago
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    Now we can find the area of triangle AOB: \(A = \dfrac{bh}{2} \) \(A = \dfrac{6 \times 3 \sqrt 3}{2} \) \(A = 9\sqrt 3\)

  73. mathstudent55
    • one year ago
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    You are correct.

  74. mathstudent55
    • one year ago
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    I'll show you that again, about the 60-60-60 triangle.

  75. anonymous
    • one year ago
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    okay

  76. mathstudent55
    • one year ago
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    You understood up to here that angle ABC is a right angle, and that angle BAC must be 60 deg.? |dw:1435340411852:dw|

  77. anonymous
    • one year ago
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    yes

  78. mathstudent55
    • one year ago
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    Now let's mark the congruent segments we have because they are radii of the circle. |dw:1435340566038:dw|

  79. mathstudent55
    • one year ago
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    Ok?

  80. anonymous
    • one year ago
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    yes yes

  81. anonymous
    • one year ago
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    =)

  82. mathstudent55
    • one year ago
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    Great. Now let's concentrate only on triangle AOB.

  83. mathstudent55
    • one year ago
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    |dw:1435340649665:dw|

  84. anonymous
    • one year ago
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    so AO and BO makes it an isosceles

  85. mathstudent55
    • one year ago
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    We have a triangle, ABO, in the figure above. Sides AO and BO are congruent. That means the opposite angles, angles A and B are congruent.

  86. mathstudent55
    • one year ago
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    Yes, triangle AOB is isosceles.

  87. mathstudent55
    • one year ago
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    |dw:1435340749447:dw|

  88. anonymous
    • one year ago
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    but we dont now what AB is it could be anything

  89. mathstudent55
    • one year ago
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    Yes, so far you are correct. We don;t know anything about AB.

  90. anonymous
    • one year ago
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    oh noooooooooooooooooooooooo so because AO is 60 then BO would be 60 so the remaining would also be 60 .....

  91. mathstudent55
    • one year ago
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    We already know that angle A measures 60, so we can conclude that angle B also measures 60. That is because angles A and B are opposite congruent sides,so they must be congruent.

  92. mathstudent55
    • one year ago
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    |dw:1435340885281:dw|

  93. anonymous
    • one year ago
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    i get it now ohhhhhhh okay wow i'm dumb that took me forever to get ...

  94. mathstudent55
    • one year ago
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    \(\Huge \bf \color{red}{BINGO!!!} \)

  95. anonymous
    • one year ago
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    Thank you so much <333333333333333333333333333333

  96. mathstudent55
    • one year ago
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    You are very welcome. I'm glad we took the extra time for you to understand it completely.

  97. anonymous
    • one year ago
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    Couldn't have done it without you thanks!!!!!! =*)

  98. mathstudent55
    • one year ago
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    You're welcome. Glad to help.

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