A triangle has corners at #(9 ,4 )#, #(7 ,5 )#, and #(3 ,6 )#. What is the area of the triangle's circumscribed circle?

1 Answer
Ratnaker Mehta
Jun 22, 2016

Reqd. Area=#212.5pi sq.unit~=667.25sq.unit.#

Explanation:

To find the reqd. area of the circumcircle of the #Delta ABC#, where #A(9,4), B(7,5), C(3,6)# , we have to first find out the radius of the circle, say #R.#

Suppose that, pt.#P(x,y)# is the circumcentre of #Delta ABC.#

Then, dist. #PA#=dist. #PB#=dist. #PC,# each #=R.#

#:. (PA)^2=(PB)^2=(PC)^2.# Using Dist. Formula, we get,

#(x-9)^2+(y-4)^2=(x-7)^2+(y-5)^2=(x-3)^2+(y-6)^2.#

#:. -18x+81-8y+16=-14x+49-10y+25=-6x+9-12y+36#

#:. -4x+2y+23=0..........(1),# [using first & second eqns.], &
#-8x+2y+29=0.............(2)#,[using second & third eqns.]

Then, #(1)-(2)# gives, #4x-6=0,# or, #x=3/2#, then by #(1), y=-17/2#

So, the circumcentre of #Delta ABC# is #P(3/2,-17/2).#

Hence, #R^2=(CP)^2=(3-3/2)^2+(6+17/2)^2=9/4+841/4=850/4=425/2,# giving the Area of Circumcircle of #DeltaABC#=#pi*R^2=425/2pi=212.5pi~=667.25#

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