A triangle has corners at #(5 ,8 )#, #(2 ,6 )#, and #(7 ,3 )#. What is the area of the triangle's circumscribed circle?

Answer 1

The area of the triangle's circumscribed circle is:
#Delta~~23.2499 ,sq.units#

Let , #triangle ABC# be the triangle with corners at

#A(5,8), B(2,6) and C(7,3) .#

Using Distance formula ,we get

#a=BC=sqrt((7-2)^2+(3-6)^2)=sqrt(25+9)=sqrt34#

#b=CA=sqrt((5-7)^2+(8-3)^2)=sqrt(4+25)=sqrt29#

#c=AB=sqrt((5-2)^2+(8-6)^2)=sqrt(9+4)=sqrt13#

Using cosine Formula ,we get

#cosB=(c^2+a^2-b^2)/(2ca)=(13+34-29)/(2sqrt13sqrt34)=18/(2sqrt442)=9/sqrt442#

We know that,

#sin^2B=1-cos^2B#

#=>sin^2B=1-9/442=433/442#

#=>sinB=(sqrt(433/442))to[because Bin(0 ^circ,180^circ)]#

Using sine formula:we get

#b/sinB=2R=>R=b/(2sinB)#

#=>R=sqrt29/(2(sqrt(433/442)))~~2.75#

So , the area of the triangle's circumscribed circle is:

#Delta=piR^2=pi*(sqrt29/(2(sqrt(433/442))))^2=pi((29xx442)/(4 xx433))#

#Delta~~23.2499 ,sq.units#

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Answer 2

To find the area of the circumscribed circle of a triangle, you first need to calculate the lengths of the triangle's sides. Then, you can use Heron's formula to find the area of the triangle. After that, you can determine the radius of the circumscribed circle using the formula:

[ R = \frac{abc}{4A} ]

Where ( a ), ( b ), and ( c ) are the lengths of the sides of the triangle, and ( A ) is the area of the triangle.

Using the given coordinates, you can calculate the lengths of the sides of the triangle using the distance formula:

[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ]

Then, you can find the area of the triangle using Heron's formula:

[ A = \sqrt{s(s - a)(s - b)(s - c)} ]

Where ( s ) is the semi-perimeter of the triangle, given by:

[ s = \frac{a + b + c}{2} ]

After finding the area of the triangle, you can plug the values into the formula for the radius of the circumscribed circle to find the answer.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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