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In the figure above, showing circle with center O and points

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In the figure above, showing circle with center O and points [#permalink] New post 28 Feb 2013, 04:23
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In the figure above, showing circle with center O and points A, B, and C on the circle, given that length (AB)= 12 and length (BC)=4\sqrt{3}, what is the area of triangle AOB?

A. 12\sqrt{3}

B. 24\sqrt{3}

C. 48

D. 48\sqrt{3}

E. 72
[Reveal] Spoiler: OA

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Last edited by Bunuel on 28 Feb 2013, 05:57, edited 1 time in total.
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Re: In the figure above, showing circle with center O and points [#permalink] New post 11 Jun 2013, 02:54
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As AC (diameter of the cirlcle) is one side of the triangle, ABC is right angle triangle and B is 90 degrees.

Area of ABS is 1/2 * 12*4*sqrt(3) = 24 Sqrt(3)
The are of AOB must be less than ABC.
in the given options only option A is less than 24 Sqrt(3)
Answer is - A
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Re: In the figure above, showing circle with center O and points [#permalink] New post 28 Feb 2013, 06:07
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In the figure above, showing circle with center O and points A, B, and C on the circle, given that length (AB)= 12 and length (BC)=4\sqrt{3}, what is the area of triangle AOB?

A. 12\sqrt{3}

B. 24\sqrt{3}

C. 48

D. 48\sqrt{3}

E. 72

Important property:
A right triangle inscribed in a circle must have its hypotenuse as the diameter of the circle. The reverse is also true: if the diameter of the circle is also the triangle’s side, then that triangle is a right triangle.

According to the above, we have that ABC is a right triangle, with a right angle at B. Thus the area of ABC is \frac{1}{2}*AB*BC=\frac{1}{2}*12*4\sqrt{3}=24\sqrt{3}.

Now, notice that since OA=OC=radius, then BO turns out to be the median of triangle ABC.

Another important property:
Each median divides the triangle into two smaller triangles which have the same area.

So, (area of AOB) = (area of COB) = 1/2 (area of ABC) = \frac{1}{2}*24\sqrt{3}=12\sqrt{3}.

Answer: A.

For more check here: math-triangles-87197.html
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Re: In the figure above, showing circle with center O and points [#permalink] New post 11 Jun 2013, 23:56
emmak wrote:
Attachment:
1.jpg
In the figure above, showing circle with center O and points A, B, and C on the circle, given that length (AB)= 12 and length (BC)=4\sqrt{3}, what is the area of triangle AOB?

A. 12\sqrt{3}

B. 24\sqrt{3}

C. 48

D. 48\sqrt{3}

E. 72


Area of ABC = 1/2*AB*BC = 1/2*12*4\sqrt{3} = 24\sqrt{3}
Area of AOB = 1/2 Area of ABC = 12\sqrt{3}

Hence, A is correct.
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Re: In the figure above, showing circle with center O and points [#permalink] New post 12 Jun 2013, 00:42
Thank you for another great geometry question, emmak. Now let's solve it properly :)

Attachment:
1.jpg
1.jpg [ 6.23 KiB | Viewed 1128 times ]

The first thing you'll notice in the figure above is that the hypothenuse (the biggest side in a triangle - in this case AC) is the diameter of the circle. Which automatically means that the ABC triangle is a right triangle. Therefore, the Pythagorean theorem become applicable.

We know the lengths of AB and BC, we can therefore deduce the length of AC by using the Pythagorean theorem.

First of all : AC^2 = AB^2 + BC^2

By plugging in the values of AB and BC, we get :

AC^2 = 12^2 + (4\sqrt{3})^2
AC^2 = 144 + 16*3 = 144 + 48 = 192
AC = \sqrt{192}

By breaking down 192 into prime factors we get : 192 = 2^6 * 3

So AC = 8\sqrt{3}

Since O is the center of the circle, then AO = OC = OB = \frac{AC}{2} = 4\sqrt{3}

Notice that the triangle AOB is an isoceles triangle (since AO = OB) and the formula for calculating the area of a triangle is : Area = \frac{base * height}{2}

The base here is AB. We need to find the height. Since AOB is an isoceles triangle, then the height will cut the base in half. So, noting H as the height of AOB, we can apply the Pythagorean theorem to deduce the value of the height :

H^2 + \frac{AB^2}{4} = OB^2

H = \sqrt{OB^2 - \frac{AB^2}{4}}

By plugging in the values of AB and OB we get :

H = \sqrt{16*3 - 6^2} = \sqrt{48 - 36} = \sqrt{12} = 2*\sqrt{3}

We can finally apply the formula to calculate the area of AOB :

Area of AOB = \frac{H*AB}{2} = 2*12*\sqrt{3}/2 = 12*\sqrt{3}

Which corresponds to answer choice A.

Hope that helped :-D
Re: In the figure above, showing circle with center O and points   [#permalink] 12 Jun 2013, 00:42
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