A triangle has corners at #(3 ,5 )#, #(4 ,9 )#, and #(7 ,6 )#. What is the area of the triangle's circumscribed circle?

Question

A triangle has corners at #(3  ,5   )#, #(4   ,9   )#, and #(7   ,6  )#.  What is the area of the triangle's circumscribed circle?

Aurora Ayala · Accepted Answer

The area of the circumscribed circle is:

#A = 5.78pi# When I do this type problem, I always shift all of the corners so that one of them is the origin. This does not change the area of the circumscribed circle and it makes one of the 3 equations (that we must write) become very simple and useful. Subtract 3 from every x coordinate and subtract 5 from every y coordinate: #(3,5) to (0,0)# #(4,9) to (1, 4)# #(7,6) to (4, 1)# Use the standard form of the equation of a circle, #(x - h)^2 + (y - k)^2 = r^2# , and the 3 shifted points to write three equations: #(0 - h)^2 + (0 - k)^2 = r^2" [1]"# #(1 - h)^2 + (4 - k)^2 = r^2" [2]"# #(4 - h)^2 + (1 - k)^2 = r^2" [3]"# Please observe that equation [1] simplifies to: #h^2 + k^2 = r^2" [4]"# This allows us to substitute #h^2 + k^2 # for #r^2# in equations [2] and [3]: #(1 - h)^2 + (4 - k)^2 = h^2 + k^2" [5]"# #(4 - h)^2 + (1 - k)^2 = h^2 + k^2" [6]"# Expand the squares, using the pattern #(a - b)^2 = a^2 - 2ab + b^2: #1 - 2h + h^2 + 16 - 8k + k^2 = h^2 + k^2" [5]"# #16 - 8h + h^2 + 1 - 2k + k^2 = h^2 + k^2" [6]"# #h^2 and k^2# are on both side of the equation, therefore, they sum to zero: #1 - 2h + 16 - 8k = 0" [7]"# #16 - 8h + 1 - 2k = 0" [8]"# collect the constant terms into a single term on the right: #-2h - 8k = -17" [7]"# #-8h - 2k = -17" [8]"# Multiply equation [8] by -4 and add to equation [7]: #30h = 51# #h = 1.7# Substitute 1.7 for h in equation [8]: #-8(1.7) - 2k = -17# #-2k = -3.4# #k = 1.7# Use equation [4] to find the value of #r^2#: #r^2 = 1.7^2 + 1.7^2# #r^2 = 5.78# The area of a circle is: #A = pir^2# The area of the circumscribed circle is: #A = 5.78pi#

Sophia Alfred · Answer

undefined To find the area of the triangle's circumscribed circle, follow these steps:

1. Calculate the lengths of the sides of the triangle using the distance formula.
2. Use Heron's formula to find the area of the triangle.
3. Find the radius of the circumscribed circle using the formula $R = \frac{abc}{4A}$, where $a$, $b$, and $c$ are the lengths of the sides of the triangle and $A$ is its area.
4. Finally, calculate the area of the circle using the formula for the area of a circle: $A = \pi R^2$.