A triangle has corners at #(9 ,4 )#, #(7 ,1 )#, and #(3 ,9 )#. What is the area of the triangle's circumscribed circle?

Answer 1

Area of circumscribed circle is #63.5496#

If the sides of a triangle are #a#, #b# and #c#, then the area of the triangle #Delta# is given by the formula
#Delta=sqrt(s(s-a)(s-b)(s-c))#, where #s=1/2(a+b+c)#
and radius of circumscribed circle is #(abc)/(4Delta)#
Hence let us find the sides of triangle formed by #(9,4)#, #(7,1)# and #(3,9)#. This will be surely distance between pair of points, which is
#a=sqrt((7-9)^2+(1-4)^2)=sqrt(4+9)=sqrt13=3.6056#
#b=sqrt((3-7)^2+(9-1)^2)=sqrt(16+64)=sqrt80=8.9443# and
#c=sqrt((3-9)^2+(9-4)^2)=sqrt(36+25)=sqrt61=7.8102#
Hence #s=1/2(3.6056+8.9443+7.8102)=1/2xx20.3601=10.1801#
and #Delta=sqrt(10.1801xx(10.1801-3.6056)xx(10.1801-8.9443)xx(10.1801-7.8102)#
= #sqrt(10.1801xx6.5745xx1.2358xx2.3699)=sqrt16.01=14.0006#

And radius of circumscribed circle is

#(3.6056xx8.9443xx7.8102)/(4xx14.0006)=4.4976#
And area of circumscribed circle is #3.1416xx(4.4976)^2=63.5496#
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Answer 2

To find the area of the circumscribed circle of a triangle, you can use the formula:

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

Where:

  • ( a ), ( b ), and ( c ) are the lengths of the sides of the triangle.
  • ( R ) is the radius of the circumscribed circle.

First, you need to find the lengths of the sides of the triangle using the distance formula:

[ d = \sqrt{{(x2 - x1)^2 + (y2 - y1)^2}} ]

Then, you can find the radius of the circumscribed circle using the formula:

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

Substituting the values of the sides and the area into the formula will give you the radius of the circumscribed circle.

Once you have the radius, you can find the area of the circle using the formula:

[ A_{\text{circle}} = \pi R^2 ]

Where ( \pi ) is approximately 3.14159.

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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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