A triangle has corners at #(9 ,7 )#, #(3 ,1 )#, and #(5 ,2 )#. What is the area of the triangle's circumscribed circle?

Question

A triangle has corners at #(9  ,7   )#, #(3   ,1   )#, and #(5   ,2  )#.  What is the area of the triangle's circumscribed circle?

James Atkins · Accepted Answer

The area of the circumscribed circle is:

#A = (205pi)/2# Shift the given points so that one of them is the origin: #(3-3,1-1)to (0,0)# #(9-3,7-1)to (6,6)# #(5-3,2-1)to (2,1)# Use the equation of a circle #(x-h)^2+(y-k)=r^2# and the 3 new points to write 3 equations: #h^2+k^2=r^2" [1]"# #(6-h)^2+(6-k)^2=r^2" [2]"# #(2-h)^2+(1-k)^2=r^2" [3]"# Substitute the left side of equation [1] into equations [2] and [3]: #(6-h)^2+(6-k)^2=h^2+k^2" [4]"# #(2-h)^2+(1-k)^2=h^2+k^2" [5]"# Expand the squares: #36-12h+h^2+36-12k+k^2=h^2+k^2" [6]"# #4-4h+h^2+1-2k+k^2=h^2+k^2" [7]"# Subtract #h^2+k^2# from both sides: #36-12h+36-12k=0" [8]"# #4-4h+1-2k=0" [9]"# Divide the first equation by -12 and multiply the second by -1: #h+k=6" [10]"# #4h+2k=5" [11]"# Multiply equation [10] by -2 and add to equation [11]: #2h = -7# #h = -7/2# Use equation [10] to solve for k: #-7/2 + k = 6# #k = 19/2# Use equation [1] to find the value of r^2: #r^2 = (-7/2)^2+(19/2)^2# #r^2 = (-7/2)^2+(19/2)^2# #r^2 = 205/2# The area of a circle is: #A = pir^2# Therefore, the area of the circumscribed circle is: #A = (205pi)/2#

Sadie Allen · Answer

9225/8

In order to find the area of the circumscribed circle, we need to find its radius. There is a formula to do this, although it is a little troublesome. The formula for the circumradius is: R= abc/4A where R is the circumradius, a,b,c are the sides of the triangle, and A is the area of the triangle. For a proof of the formula, go to this site: https://tutor.hix.ai So we need to find the area of the triangle and its side lengths. To find the area of the triangle, we use the shoelace formula. If you don't know what that is, go to: https://tutor.hix.ai (you may also learn some other methods to calculate the area of a triangle) So A = #1/2|9*1-3*7+3*2-1*5+5*7-2*9|# = #1/2|-12+1+17|# = #1/2*6# = 3 Next, find each of the side lengths using the distance formula: a = #sqrt((5-3)^2+(2-1)^2) = sqrt(4+1) = sqrt5# b = #sqrt((9-5)^2+(7-2)^2) = sqrt(16+25) = sqrt41# c = #sqrt((9-3)^2+(7-1)^2) = sqrt(36+36) = sqrt72# Not nice numbers but still. abc = #sqrt(5*41*72) = sqrt(5*2952) = sqrt14760# R = #sqrt(14760)/12# A (circle) = #pi*R^2# = #pi* 14760/144 = 9225/8# Hope that helps!

Aurora Ayala · Answer

undefined To find the area of the circumscribed circle of a triangle, you need to know the lengths of its sides or the coordinates of its vertices. Given the coordinates of the vertices, you can calculate the lengths of the sides using the distance formula. Then, you can use these side lengths to find the radius of the circumscribed circle using the circumradius formula. Finally, you can calculate the area of the circle using the formula for the area of a circle, A = πr^2, where r is the radius.

Here's the step-by-step process:

1. Calculate the lengths of the sides of the triangle using the distance formula:
   - Length of side AB = √((x2 - x1)^2 + (y2 - y1)^2)
   - Length of side BC = √((x3 - x2)^2 + (y3 - y2)^2)
   - Length of side AC = √((x3 - x1)^2 + (y3 - y1)^2)

2. Find the semi-perimeter of the triangle:
   - Semi-perimeter = (Length of side AB + Length of side BC + Length of side AC) / 2

3. Calculate the radius of the circumscribed circle using the circumradius formula:
   - Circumradius (R) = (Length of side AB * Length of side BC * Length of side AC) / (4 * Triangle Area)

4. Calculate the area of the circumscribed circle using the formula for the area of a circle:
   - Area of the circumscribed circle = π * (Circumradius)^2

By following these steps, you can find the area of the triangle's circumscribed circle using the given coordinates of its vertices.