An isosceles triangle has sides A, B, and C with sides B and C being equal in length. If side A goes from #(7 ,1 )# to #(2 ,9 )# and the triangle's area is #18 #, what are the possible coordinates of the triangle's third corner?

1 Answer

There are two possible points at #color(blue)((1377/178"," 625/89))# and
#color(red)((225/178"," 265/89))#

Explanation:

The solution makes use of the formula for the area of a triangle

Area #=1/2[(x_1, x_2, x_3, x_1),(y_1, y_2, y_3, y_1)]#

positive area for counterclockwise arrangement of points
negative area for clockwise arrangements of points

Let (x_1, y_1) be the unknown point and Let #P_2(2, 9)# and #P_3(7, 1)# and P_1(x_1, y_1) and Area#=18#

Area #=1/2[(x_1, x_2, x_3, x_1),(y_1, y_2, y_3, y_1)]#

#+18=1/2[(x_1, 2, 7, x_1),(y_1, 9, 1, y_1)]#

#18(2)=(9x_1+2+7y_1-2y_1-63-x_1)#

#36=8x_1+5y_1-61#

#8x_1+5y_1=97" "#first equation

For the equal sides because the triangle is isosceles, we have

#(x_1-2)^2+(y_1-9)^2=(x_1-7)^2+(y_1-1)^2#
#x_1^2-4x_1+4+y_1^2-18y_1+81=x_1^2-14x_1+49+y_1^2-2y_1+1#
Simplify
#-4x_1+4-18y_1+81=-14x_1+49-2y_1+1#
#10x_1-16y_1+85-50=0#
#10x_1-16y_1+35=0" "#second equation

Simultaneous solution using first and second equations results to
#color(red)((x_1, y_1)=(1377/178, 625/89))#

There is another point which may be solved by using the negative area

Area #=1/2[(x_1, x_2, x_3, x_1),(y_1, y_2, y_3, y_1)]#
#-18=1/2[(x_1, 2, 7, x_1),(y_1, 9, 1, y_1)]#

#-18(2)=(9x_1+2+7y_1-2y_1-63-x_1)#

#-36=8x_1+5y_1-61#

#8x_1+5y_1=25" "#third equation

Simultaneous solution using second and third equations results to
#color(red)((x_1, y_1)=(225/178, 265/89))#

God bless....I hope the explanation is useful.