A triangle has corners at #(3 ,8 )#, #(7 ,9 )#, and #(4 ,6 )#. What is the area of the triangle's circumscribed circle?

Question

A triangle has corners at #(3  ,8   )#, #(7   ,9   )#, and #(4   ,6  )#.  What is the area of the triangle's circumscribed circle?

Ella Armstrong · Accepted Answer

#Area = (170pi)/36#

The standard Cartesian form of the equation of a circle is:

#(x - h)^2 + (y - k)^2 = r^2" [1]"#

where #(x, y)# is any point on the circle, #(h, k)# is the center point and r is the radius.

Use equation [1] and the 3 points to write 3 equations:

#(3 - h)^2 + (8 - k)^2 = r^2" [2]"# #(7 - h)^2 + (9 - k)^2 = r^2" [3]"# #(4 - h)^2 + (6 - k)^2 = r^2" [4]"#

Expand the squares, using the pattern #(a - b) = a^2 - 2ab + b^2#:

#9 - 6h + h^2 + 64 - 16k + k^2 = r^2" [5]"# #49 - 14h + h^2 + 81 - 18k + k^2 = r^2" [6]"# #16 - 8h + h^2 + 36 - 12k + k^2 = r^2" [7]"#

Set the left side of equation [5] equal to the left side of equation [6]:

#9 - 6h + h^2 + 64 - 16k + k^2 = 49 - 14h + h^2 + 81 - 18k + k^2" [8]"#

Set the left side of equation [7] equal to the left side of equation [6]:

#16 - 8h + h^2 + 36 - 12k + k^2 = 49 - 14h + h^2 + 81 - 18k + k^2" [9]"#

There #h^2 and k^2# terms on both sides of equations [8] and [9], therefore, they sum to zero:

#9 - 6h + 64 - 16k = 49 - 14h + 81 - 18k" [10]"# #16 - 8h + 36 - 12k = 49 - 14h + 81 - 18k" [11]"#

Collect all of the constant terms into a single term on the right:

#- 6h - 16k = - 14h - 18k + 57" [12]"# #- 8h - 12k = - 14h - 18k + 78" [13]"#

Collect the h terms into a single term on the left:

#8h - 16k = - 18k + 57" [14]"# #6h - 12k = - 18k + 78" [15]"#

Collect the k terms into a single term on the left:

#8h + 2k = 57" [16]"# #6h + 6k = 78" [17]"#

Multiply equation [17] by #-1/3# and add to equation [16]:

#6h = 31#

#h = 31/6#

Substitute #31/6# for h into equation [17]:

#6(31/6) + 6k = 78" [17]"#

#k = 47/6#

Substitute the values for h and k into equation [3}:

#(7 - 31/6)^2 + (9 - 47/6)^2 = r^2#

#r^2 = (42/6 - 31/6)^2 + (54 - 47/6)^2#

#r^2 = 170/36#

#Area = pir^2#

#Area = (170pi)/36#

Carson Arnold · Answer

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

Area = (abc) / (4R)

Where:
a, b, and c are the lengths of the sides of the triangle, and
R is the radius of the circumscribed circle.

To find the lengths of the sides of the triangle, you can use the distance formula between the given points.

Once you have the lengths of the sides, you can use Heron's formula to find the area of the triangle. Then, you can use the formula for the radius of the circumscribed circle:

R = (abc) / (4Area)

Substitute the values you've found into this formula to calculate the radius of the circumscribed circle. Once you have the radius, you can find the area of the circle using the formula for the area of a circle:

Area_circle = πR^2

Substitute the radius you found into this formula to get the area of the circumscribed circle.