An isosceles triangle has sides A, B, and C with sides B and C being equal in length. If side A goes from #(7 ,3 )# to #(8 ,9 )# and the triangle's area is #45 #, what are the possible coordinates of the triangle's third corner?

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Answer 1 · 2016-11-16T00:50:43

#(1635/74, 264/74)# or #(-525/74, 624/74)#

Let's find the length of side A:

#A = sqrt((8 -7)^2 + (9 - 3)^2)#

#A = sqrt(37)#

#Area = 45 = 1/2sqrt(37)h#

#h = 90sqrt(37)/37#

The height must go through the point #(7.5, 6)#

The slope of the line for the height is:

#m = (7 - 8)/(9 - 3) = -1/6#

The equation of the line for height is:

#y = -1/6(x - 7.5) + 6#

Using the distance formula:

#h = sqrt((x - 7.5)^2 + (y - 6)^2)#

#90^2/37 = (x - 7.5)^2 +(x - 7.5)^2/36#

#90^2/37 = (36(x - 7.5)^2)/36 +(x - 7.5)^2/36#

#90^2/37 = 37(x - 7.5)^2/36#

#90^2/37^2 = (x - 7.5)^2/36#

x - 7.5 = +-6(90/37)

x = 1635/74 and x = -525/74

y = 264/74 and y = 624/74