A triangle has corners at #(5 ,1 )#, #(2 ,7 )#, and #(7 ,3 )#. What is the area of the triangle's circumscribed circle?

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Answer 1 · 2018-06-10T12:00:29

Area of triangle's circumscribed circle is #35.78# sq.unit.

The three corners are #A (5,1) B (2,7) and C (7,3)#

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 ((5-2)^2+(1-7)^2) ~~ 6.708#unit

Side #BC= sqrt ((2-7)^2+(7-3)^2) ~~6.403#unit

Side #CA= sqrt ((7-5)^2+(3-1)^2) ~~ 2.828#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(5(7−3)+2(3−1)+7(1−7))|# or

#A_t = |1/2(20+4-42)| = | -9.0| =9.0# sq.unit.

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

#R=(sqrt(45)*sqrt(41)*sqrt(9))/(4*9) ~~ 3.375# unit

Area of circumscribed circle is #A_c=pi*R^2=pi*3.375^2~~35.78#

sq.unit [Ans]