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 ,5 )# and the triangle's area is #32 #, what are the possible coordinates of the triangle's third corner?

1 Answer

Possible coordinates are#(881/82, 443/41)# and #(-143/82, -197/41)#

Explanation:

We use the formula for the area A

#A=1/2[(x_a, x_b, x_c, x_a),(y_a, y_b, y_c, y_a)]#

#A=1/2[x_a*y_b+x_b*y_c+x_c*y_a-x_b*y_a-x_c*y_b-x_a*y_c]#

We have to consider two points on each side of side #a#

For the first point #A#

#A=1/2[(x_a, x_b, x_c, x_a),(y_a, y_b, y_c, y_a)]#

#A=1/2[x_a*y_b+x_b*y_c+x_c*y_a-x_b*y_a-x_c*y_b-x_a*y_c]#

#32=1/2[(x_a, 2, 7, x_a),(y_a, 5, 1, y_a)]#

#32=1/2[5x_a+2+7y_a-2y_a-35-x_a]#

#64-2+35=4x_a+5y_a#

#97=4x_a+5y_a#

This is an isosceles triangle so that the third vertex #A(x_a, y_a)# passes through the perpendicular bisector of side #a#

Perpendicular bisector equation is passing through the midpoint of side #a# which #M(9/2, 3)# with slope #m_p=5/4#

#y-3=5/4(x-9/2)#

#10x-8y=21#

Also this is

#10x_a-8y_a=21#

Solving Vertex #A# with equation #97=4x_a+5y_a#

#4x_a+5y_a=97#
#10x_a-8y_a=21#

Simultaneous solution results to

#color(red)(A(x_a, y_a)=(881/82, 443/41))#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

For the second point #A# on the other side of side #a#

#A=1/2[(x_a, x_c, x_b, x_a),(y_a, y_c, y_b, y_a)]#

#A=1/2[x_a*y_b+x_b*y_c+x_c*y_a-x_b*y_a-x_c*y_b-x_a*y_c]#

#32=1/2[(x_a, 7, 2, x_a),(y_a, 1, 5, y_a)]#

#32=1/2[5x_a+2+7y_a-2y_a-35-x_a]#

#64+2-35=-4x_a-5y_a#

#4x_a+5y_a=-31#

This is an isosceles triangle so that the other third vertex #A(x_a, y_a)# passes through the perpendicular bisector of side #a#

Perpendicular bisector equation is passing through the midpoint of side #a# which #M(9/2, 3)# with slope #m_p=5/4#

#y-3=5/4(x-9/2)#

#10x-8y=21#

Also this is

#10x_a-8y_a=21#

Solving the other Vertex #A# with equation #4x_a+5y_a=-31#

#4x_a+5y_a=-31#
#10x_a-8y_a=21#

Simultaneous solution results to

#color(red)(A(x_a, y_a)=(-143/82, -197/41))#

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