A triangle has corners at #(9 ,7 )#, #(2 ,1 )#, and #(5 ,2 )#. What is the area of the triangle's circumscribed circle?

Question

A triangle has corners at #(9  ,7   )#, #(2   ,1   )#, and #(5   ,2  )#.  What is the area of the triangle's circumscribed circle?

Aurora Ayala · Accepted Answer

#"Details are shown below. Please check my math."#

The corner coordinates of the ABC triangle are on the circumference circle.
the first step is to find the edge lengths of triangle a, b, c.
We can find the distance between two known coordinates by using the following formula.
#P_1(x_1,y_1)" , "P_2(x_2,y_2)#
#l=sqrt((x_2-x_1)^2+(y_2-y_1)^2)#
1. The length of a side:
  #a=sqrt((5-2)^2+(2-1)^2)=sqrt(3^2+1^2)=sqrt(9+1)=sqrt(10)" units"#
2. The length of b side:
  #b=sqrt((9-5)^2+(7-2)^2)=sqrt(4^2+5^2)=sqrt(16+25)=sqrt(41)" units"#
3. The length of c side:
  #c=sqrt((9-2)^2+(7-1)^2)=sqrt(7^2+6^2)=sqrt(49+36)=sqrt(85)" units"#
  -In the second step, we can calculate the area of the triangle known as corner coordinates.
  
  #A(x_1,y_1)" , "B(x_2,y_2)" , "C=(x_3,y_3)#
  #A(9,7)" , "B(2,1)" , "C(5,2)#
  
  #"triangle's area="1/2*|9*1+2*2+5*7-2*7-5*1-9*2 |#
  #"triangle's area="1/2*|9+4+35-14-5-18 |#
  #"triangle's area="1/2*|9+4+35-14-5-18 |=5.5" units"^2#
  - now we can use the formula given below.
    #area(ABC)=(a*b*c)/(4*r)#
    #5.5=(sqrt (10)*sqrt(41)*sqrt(85))/(4*r)#
    #22r=sqrt(10*41*85)#
    #r=(sqrt(10*41*85))/22#
    #r=(sqrt(34850))/22#
    #r=8.49" units"#
    - the area of the triangle's circumscribed circle:
      #area=pi*r^2#
      #area=3.14*(8.49)^2#
      #area=226.33" units"^2#

Parker Allen · Answer

undefined To find the area of the triangle's circumscribed circle, we first need to find the circumradius (the radius of the circle that circumscribes the triangle). The circumradius can be found using the formula:

$$ R = \frac{abc}{4A} $$

where $ R $ is the circumradius, $ a $, $ b $, and $ c $ are the lengths of the sides of the triangle, and $ A $ is the area of the triangle. We can find the lengths of the sides of the triangle using the distance formula between the given points. Then, we can use Heron's formula to find the area of the triangle. Finally, we can substitute the values into the formula for the circumradius and calculate it. Once we have the circumradius, we can find the area of the circumscribed circle using the formula for the area of a circle:

$$ A_{circle} = \pi R^2 $$

Substituting the value of $ R $ into this formula will give us the area of the circle.

Let's go through the steps:

1. Calculate the lengths of the sides of the triangle using the distance formula.
2. Calculate the area of the triangle using Heron's formula.
3. Substitute the values into the formula for the circumradius and calculate it.
4. Substitute the value of the circumradius into the formula for the area of the circle and calculate the area.

After performing these calculations, we'll have the area of the triangle's circumscribed circle.