An isosceles triangle has sides A, B, and C with sides B and C being equal in length. If side A goes from #(2 ,5 )# to #(8 ,7 )# and the triangle's area is #16 #, what are the possible coordinates of the triangle's third corner?

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Answer 1 · 2018-04-12T06:10:46

The coordinates of A is #A(0.4545,9.8182)#

Let the coordinates of A, B and C be #(x,y)#, #(2,5)# and #(8,7)# respectively.

Given that, #AB=AC#

#rarrAB^2=AC^2#

#rarr(2-x)^2+(5-y)^2=(8-x)^2+(7-y)^2#

#rarr4-4x+cancel(x^2)+25-10y+cancel(y^2)=64-16x+cancel(x^2)+49-14ycancel(+y^2)#

#rarr3x+2y=21#......[1]

Also, Area of #DeltaABC=16#

#rarr1/2|(x,y,1),(2,5,1),(8,7,1)|=16#

#rarrx(5-7)-y(2-8)+1(14-40)=32#

#rarr-2x+6y-26=32#

#rarrx-3y=4#.....[2]

Solving [1] and [2], we get,

#x=0.4545 and y=9.8182#

So, the coordinates of A is #(0.4545,9.8182)#.