A triangle has corners at #(9 ,5 )#, #(2 ,1 )#, and #(3 ,6 )#. What is the area of the triangle's circumscribed circle?

Question

A triangle has corners at #(9  ,5   )#, #(2   ,1   )#, and #(3   ,6  )#.  What is the area of the triangle's circumscribed circle?

Noah Atkinson · Accepted Answer

Shift ALL the points so that one is the origin. Use the standard Cartesian form for the equation of a circle and the new points to write 3 equations. Use the 3 equations to solve for #r^2#.

Shift all 3 points so that one of them is the origin: #(2,1) - (2,1) = (0,0)# #(9,5) - (2,1) = (7,4)# #(3,6) - (2,1) = (1,5)# This is the standard Cartesian form for the equation of a circle: #(x - h)^2 + (y - k)^2 = r^2" [1]"# Use the new points to write 3 equations: #(0 - h)^2 + (0 - k)^2 = r^2" [2]"# #(7 - h)^2 + (4 - k)^2 = r^2" [3]"# #(1 - h)^2 + (5 - k)^2 = r^2" [4]"# Expand the squares: #h^2 + k^2 = r^2" [5]"# #49 - 14h + h^2 + 16 - 8k + k^2 = r^2" [6]"# #1 - 2h + h^2 + 25 - 10k + k^2 = r^2" [7]"# Subtract equation [6] from equation [5] and equation [7] from equation [5]: #-49 + 14h - 16 + 8k = 0" [8]"# #-1 + 2h - 25 + 10k = 0" [9]"# Collect the constant terms into a single term on the right: #14h + 8k = 65" [10]"# #2h + 10k = 26" [11]"# Multiply equation [11] by -7 and add to equation [10]: #-62k = -117# #k = 117/62# Substitute the value for k into equation [11] and solve for h: #2h + 1170/62 = 26" [11]"# #h = 221/62# Use equation [5] to solve for #r^2#: #r^2 = h^2 + k^2" [5]"# #r^2 = (221/62)^2 + (117/62)^2# #r^2 = 62530/3844 = 31265/1922# The area of the circle is: #A = (31265pi)/1922#

Sadie Allen · Answer

undefined To find the area of the triangle's circumscribed circle, you can use the formula $ A = \frac{abc}{4R} $, where $ A $ is the area of the triangle, $ a $, $ b $, and $ c $ are the lengths of the sides of the triangle, and $ R $ is the radius of the circumscribed circle.

First, you need to calculate the lengths of the sides of the triangle using the given coordinates:

- Side $ a $ is the distance between (9, 5) and (2, 1).
- Side $ b $ is the distance between (2, 1) and (3, 6).
- Side $ c $ is the distance between (3, 6) and (9, 5).

Next, find the semi-perimeter of the triangle, $ s = \frac{a + b + c}{2} $.

Then, use Heron's formula to calculate the area of the triangle:

$$ A = \sqrt{s(s - a)(s - b)(s - c)} $$

After finding the area of the triangle, you can determine the radius of the circumscribed circle using the formula $ R = \frac{abc}{4A} $.

Finally, calculate the area of the circumscribed circle using the formula for the area of a circle $ A_{circle} = \pi R^2 $.