A triangle has corners at #(3 ,8 )#, #(5 ,9 )#, and #(8 ,5 )#. What is the area of the triangle's circumscribed circle?

2 Answers
Jun 1, 2018

Area of the triangle's circumscribed circle#= (53pi)/9# units.

Explanation:

First of all, we have to find out circumcenter (G) of the triangle.

#G=(3+5+8)/3,(8+9+5)/3 #

#G=(16)/3,(22)/3 #

For area of the triangle's circumscribed circle, we have to calculate radius of the circumscribed circle. And, it is equal to the distance between G and any of the vertices of the triangle.

#R= sqrt((16/3 -3)^2+(22/3 -8)^2#

#R= sqrt(((16-9)/3)^2+((22-24)/3)^2#

#R= sqrt(((7)/3)^2+((-2)/3)^2#

#R= sqrt(49/9+4/9#

#R= sqrt(53/9#

#R= sqrt(53)/3# units

Area of the triangle's circumscribed circle#= piR^2#

Area of the triangle's circumscribed circle#= pixx53/9#

Area of the triangle's circumscribed circle#= (53pi)/9# units.

Jun 5, 2018

#{ 2125 pi }/242 #

Explanation:

There are no square roots needed. Please contrast this with the other answer, typical good answer.

Archimedes' Theorem says for a triangle with sides #a,b,c:#

# 16 \ text{area}^2 = 4a^2b^2-(c^2-a^2-b^2)^2#

The radius of the circumcircle equals the product of the sides of the triangle divided by four times the area of the triangle. This is more useful squared:

# r^2 = {a^2 b^2 c^2}/{16 \ text{area}^2} = {a^2 b^2 c^2}/{ 4a^2b^2-(c^2-a^2-b^2)^2} #

#a^2,b^2 and c^2# are available directly from #(3,8),(5,9),(8,5)#

# a^2 = (3-5)^2+(8-9)^2=5#

#b^2 =(5-8)^2+(9-5)^2=25#

#c^2=(8-3)^2+(5-8)^2=34 #

# text{circumcircle area} = pi r^2 = pi {(5)(25)(34) }/{ 4(5)(25) - (34-5-25)^2 } ={2125 pi }/242 #

It appears me or the other answer is wrong.

Check: Alpha # quad # I'm right #quad sqrt#