First we let #C(x, y)# be the unknown 3rd corner of the triangle.
Also Let corners #A(7, 2)# and #B(3, 6)#
We set the equation using sides by distance formula
#a=b#
#sqrt((x_c-3)^2+(y_c-6)^2)=sqrt((x_c-7)^2+(y_c-2)^2)#
simplify to obtain
#x_c-y_c=1" " "#first equation
Use now the matrix formula for Area:
#Area=1/2((x_a,x_b,x_c,x_a),(y_a,y_b,y_c,y_a))=#
#=1/2(x_ay_b+x_by_c+x_cy_a-x_by_a-x_cy_b-x_ay_c)#
#Area=1/2((7,3,x_c,7),(2,6,y_c,2))=#
#Area=1/2*(42+3y_c+2x_c-6-6x_c-7y_c)#
#Area=6# this is given
We now have the equation
#6=1/2*(42+3y_c+2x_c-6-6x_c-7y_c)#
#12=-4x_c-4y_c+36#
#x_c+y_c=6" " " #second equation
Solving simultaneously the system
#x_c-y_c=1#
#x_c+y_c=6#
#x_c=7/2# and #y_c=5/2#
We can now solve for the lengths of sides #a# and #b#
#a=b=sqrt((x_b-x_c)^2+(y_b-y_c)^2)#
#a=b=sqrt((3-7/2)^2+(6-5/2)^2)#
#a=b=5/2sqrt(2)=3.5355339" " "#units
compute side #c#:
#c=sqrt((x_a-x_b)^2+(y_a-y_b)^2)#
#c=sqrt((7-3)^2+(2-6)^2)#
#c=sqrt(2(16))#
#c=4sqrt2=5.6568542#
