Question

Two corners of an isosceles triangle are at `(2 ,5 )` and `(9 ,8 )`. If the triangle's area is `6`, what are the lengths of the triangle's sides?

Answer 1

1.) `sqrt58`,`" " "sqrt58`,`" " "2sqrt(29+sqrt(805))`
2.) `sqrt58=7.615`,`" " "sqrt58=7.615`,`" " "2sqrt(29-sqrt(805))=1.58427`
3.)`sqrt58=7.615`,`sqrt(57130)/58=4.12101`,`sqrt(57130)/58=4.12101`

Explanation:

The possible triangles are when the given distance `sqrt58` are the two equal sides

triangle 1: the coordinates are `(2,5), (9,8), ((279+7sqrt805)/29,(190+3sqrt805)/29)`

triangle 2: the coordinates are `(2, 5), (9, 8), ((279-7sqrt805)/29,(190-3sqrt805)/29)`

For the 3rd triangle, `sqrt58` is the 3rd side and is not congruent to any side.
triangle 3: the coordinates are `(2, 5), (9, 8), (355/58, 293/58)`

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