1. anonymous

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

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3. misssunshinexxoxo

Do you know how to find the area of a triangle?

4. anonymous

yes

5. anonymous

bh/2

6. anonymous

help someone...

7. anonymous

@misty1212

8. mathstudent55

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9. mathstudent55

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10. mathstudent55

Do you understand so far?

11. anonymous

wait how do we know it's 60?

12. anonymous

what about the triangle on the right side what do those angles equal to?

13. mathstudent55

I'll show you each angle I found, one at a time, and will explain.

14. anonymous

okay

15. mathstudent55

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

okay

17. mathstudent55

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

That makes angle ABC 90 degrees, and triangle ABC a right triangle.

19. mathstudent55

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20. anonymous

right i remember all inscribed angles are 90 degrees

21. mathstudent55

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

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23. anonymous

30 60 90 yea

24. mathstudent55

Correct.

25. mathstudent55

Now look at the congruent sides of triangle AOB.

26. mathstudent55

|dw:1435338598897:dw| Triangle AOB is isosceles. Opp angles to the congruent sides are congruent.

27. mathstudent55

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28. mathstudent55

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

sorry i lost you

30. mathstudent55

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31. anonymous

Triangle AOB is isosceles. Opp angles to the congruent sides are congruent.

32. anonymous

what do u mean by that

33. mathstudent55

Did you understand that angle A of triangle AOB is 60 deg?

34. anonymous

yes

35. mathstudent55

There is a theorem that states: If a triangle has two congruent sides, then the angles opposite those sides are congruent.

36. mathstudent55

That means, in an isosceles triangle, the base angles are congruent.

37. anonymous

hmmmm.... okay

38. mathstudent55

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39. anonymous

yes angles B and C are equal

40. anonymous

but the A is different....

41. anonymous

so how does it relate

42. mathstudent55

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

That means the base angles, BAO and ABO are congruent angles.

44. mathstudent55

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45. mathstudent55

We now know that angle ABO also measures 60 deg. |dw:1435339121331:dw|

46. mathstudent55

Once two angles of a triangle are known, you can find the third angle. 60 + 60 + m<AOB = 180 m<AOB = 60

47. mathstudent55

Now we know that triangle AOB is also equilateral in addition to being isosceles.

48. anonymous

Cant it be this? |dw:1435339216515:dw|

49. anonymous

then it wouldn't be 60 60 60

50. mathstudent55

According to the given info, it must be equilateral.

51. anonymous

okay thank you

52. anonymous

im gonna have to look over this a few times im still having a hard time understanding why its all 60

53. mathstudent55

This is not the end of the problem. We still need to find the area of triangle ABO.

54. mathstudent55

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

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

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

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58. mathstudent55

Now the problem is simply to find the area of an equilateral triangle whose side has a length of 6.

59. anonymous

so bh/2 right?

60. mathstudent55

Yes, but we need to find the height. Let's just look at the triangle now.

61. anonymous

6 would be the opposite of 30 degrees

62. anonymous

and height is given as $6\sqrt{3}$

63. mathstudent55

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64. anonymous

ohhhhh yeaaaa

65. mathstudent55

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66. mathstudent55

Inside triangle AOB, we have two 30-60-90 triangles. |dw:1435340060713:dw|

67. mathstudent55

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

so it would be $9\sqrt{3}$

69. mathstudent55

Remember, the long leg is $$\sqrt 3$$ times longer than the short leg. |dw:1435340159438:dw|

70. mathstudent55

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71. anonymous

i understand everything except how it's a 60-60-60- triangle

72. mathstudent55

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

You are correct.

74. mathstudent55

I'll show you that again, about the 60-60-60 triangle.

75. anonymous

okay

76. mathstudent55

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

yes

78. mathstudent55

Now let's mark the congruent segments we have because they are radii of the circle. |dw:1435340566038:dw|

79. mathstudent55

Ok?

80. anonymous

yes yes

81. anonymous

=)

82. mathstudent55

Great. Now let's concentrate only on triangle AOB.

83. mathstudent55

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84. anonymous

so AO and BO makes it an isosceles

85. mathstudent55

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

Yes, triangle AOB is isosceles.

87. mathstudent55

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88. anonymous

but we dont now what AB is it could be anything

89. mathstudent55

Yes, so far you are correct. We don;t know anything about AB.

90. anonymous

oh noooooooooooooooooooooooo so because AO is 60 then BO would be 60 so the remaining would also be 60 .....

91. mathstudent55

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

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93. anonymous

i get it now ohhhhhhh okay wow i'm dumb that took me forever to get ...

94. mathstudent55

$$\Huge \bf \color{red}{BINGO!!!}$$

95. anonymous

Thank you so much <333333333333333333333333333333

96. mathstudent55

You are very welcome. I'm glad we took the extra time for you to understand it completely.

97. anonymous

Couldn't have done it without you thanks!!!!!! =*)

98. mathstudent55