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

2 Answers
Jun 7, 2017

The area of the circumscribed circle is #=78.54#

Explanation:

To calculate the area of the circle, we must calculate the radius #r# of the circle

Let the center of the circle be #O=(a,b)#

Then,

#(4-a)^2+(7-b)^2=r^2#.......#(1)#

#(3-a)^2+(4-b)^2=r^2#..........#(2)#

#(8-a)^2+(9-b)^2=r^2#.........#(3)#

We have #3# equations with #3# unknowns

From #(1)# and #(2)#, we get

#16-8a+a^2+49-14b+b^2=9-6a+a^2+16-8b+b^2#

#2a+6b=40#

#a+3b=20#.............#(4)#

From #(2)# and #(3)#, we get

#9-6a+a^2+16-8b+b^2=64-16a+a^2+81-18b+b^2#

#10a+10b=120#

#a+b=12#..............#(5)#

From equations #(4)# and #(5)#, we get

#12-b+3b=20#

#2b=8#

#b=8/2=4#

#a=12-b=12-4=8#

The center of the circle is #=(8,4)#

#r^2=(4-a)^2+(7-b)^2=(4-8)^2+(7-4)^2#

#=16+9#

#=25#

The area of the circle is

#A=pi*r^2=25*pi=78.54#

Jun 29, 2018

#(4-3)^2+(7-4)^2=10#

#(8-3)^2+(9-4)^2=50#

#(8-4)^2+(9-7)^2=20#

#pi r^2 = {pi (10)(50)(20) }/{ 4(10)(50) - (20-10-50)^2} = 25 pi #

Explanation:

Here's the shortcut.

The circumcircle is just the circle through the three vertices; the triangle almost doesn't matter. Except, miraculously, the circumradius #r# equals the product of the triangle sides #a,b,c# divided by four times the triangle's area #A#.

#r = {abc}/{4A}#

It's much more useful squared, and we're looking for #pi r^2# anyway.

#pi r^2 = {pi a^2 b^2 c^2}/{16A^2}#

The coordinates give the squared distances easily. Archimedes' Theorem relates the squared distances to the triangle area:

# 16A^2 = 4a^2b^2 - (c^2-a^2-b^2)^2#

So,

#pi r^2 = {pi a^2 b^2 c^2}/{ 4a^2b^2 - (c^2-a^2-b^2)^2}#

We form the squared distances from pairs of points #(4,7),(3,4), ( 8,9) #

#a^2=(4-3)^2+(7-4)^2=10#

#b^2=(8-3)^2+(9-4)^2=50#

#c^2=(8-4)^2+(9-7)^2=20#

#pi r^2 = {pi (10)(50)(20) }/{ 4(10)(50) - (20-10-50)^2} = 25 pi #