A triangle has corners at #(4 ,6 )#, #(8 ,9 )#, and #(3 ,5 )#. What is the area of the triangle's circumscribed circle?

2 Answers
Jun 2, 2018

Area of circumscribed circle is #1610.07# sq.unit .

Explanation:

The three corners are #A (4,6) B (8,9) and C (3,5)#

Distance between two points #(x_1,y_1) and (x_2,y_2)# is

#D= sqrt ((x_1-x_2)^2+(y_1-y_2)^2#

Side #AB= sqrt ((4-8)^2+(6-9)^2)=sqrt(25) = 5.0# unit

Side #BC= sqrt ((8-3)^2+(9-5)^2)=sqrt(41) ~~6.40#unit

Side #CA= sqrt ((3-4)^2+(5-6)^2)=sqrt(2) ~~ 1.41#unit

Area of Triangle is

#A_t = |1/2(x_1(y_2−y_3)+x_2(y_3−y_1)+x_3(y_1−y_2))|#

#A_t = |1/2(4(9−5)+8(5−6)+3(6−9))|# or

#A_t = |1/2(16-8-9)| = 1/2*1 =0.5# sq.unit.

Radius of circumscribed circle is #R=(AB*BC*CA)/(4*A_t)# or

#R=(5*sqrt(41)*sqrt(2))/(4*0.5) ~~ 22.64# unit

Area of circumscribed circle is #A_c=pi*R^2=pi*22.64^2#

#~~1610.07#sq.unit [Ans]

Jun 5, 2018

#{1025 pi }/2 #

Explanation:

Allow me to illustrate the proper method; no square roots needed. Please contrast this with the other answer, which is a 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} #

That all we need to do our problem. #(4,6),(8,9),(3,5)#

# a^2 = (4-8)^2+(6-9)^2=25#

#b^2 = (8-3)^2+(9-5)^2=41#

#c^2=(4-3)^2+(6-5)^2=2#

# text{circumcircle area} = pi r^2 = pi {25(41)(2)}/{4(25)(2)-(41-25-2)^2 } = {1025 pi }/2 #