A triangle has corners at #(8 ,7 )#, #(2 ,1 )#, and #(5 ,6 )#. What is the area of the triangle's circumscribed circle?

Answer 1

The area of the circle is #=133.5u^2#

To calculate the area of the circle, we must calculate the radius #r# of the circle
Let the center of the circle be #O=(a,b)#

Then,

#(8-a)^2+(7-b)^2=r^2#.......#(1)#
#(2-a)^2+(1-b)^2=r^2#..........#(2)#
#(5-a)^2+(6-b)^2=r^2#.........#(3)#
We have #3# equations with #3# unknowns
From #(1)# and #(2)#, we get
#64-16a+a^2+49-14b+b^2=4-4a+a^2+1-2b+b^2#
#12a+12b=108#
#a+b=9#.............#(4)#
From #(2)# and #(3)#, we get
#4-4a+a^2+1-2b+b^2=25-10a+a^2+36-12b+b^2#
#6a+10b=56#
#3a+5b=28#..............#(5)#
From equations #(4)# and #(5)#, we get
#3(9-b)+5b=28#
#27-3b+5b=28#
#2b=1#, #=>#, #b=1/2#
#a=9-1/2#, #=>#, #a=17/2#
The center of the circle is #=(17/2,1/2)#
#r^2=(2-17/2)^2+(1-1/2)^2=(-13/2)^2+(1/2)^2#
#=170/4#
#=85/2#

The area of the circle is

#A=pi*r^2=pi*85/2=133.5#
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Answer 2

The area of the triangle's circumscribed circle can be calculated using the formula:

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

Where:

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

First, calculate the side lengths of the triangle using the distance formula:

[a = \sqrt{(8 - 2)^2 + (7 - 1)^2}] [b = \sqrt{(2 - 5)^2 + (1 - 6)^2}] [c = \sqrt{(5 - 8)^2 + (6 - 7)^2}]

Then, calculate the semiperimeter of the triangle:

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

Next, compute the radius of the circumscribed circle using Heron's formula:

[A_{\text{triangle}} = \sqrt{s(s-a)(s-b)(s-c)}]

Finally, use the formula for the area of the circumscribed circle:

[A = \frac{abc}{4A_{\text{triangle}}}]

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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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