Question

Find FL if H is the circumcenter of EFG,

Answer 1

The artistic crop isn't helpful; it cuts off some vertex names.

The circumcenter H is the meet of the perpendicular bisectors of the sides, helpfully drawn.  We have right triangle ELH, right angle L, so

EH² = HL² + EL²

EL = √(EH² - HL²) = √(5.06²-2.74²) ≈ 4.2539393507665339

Since HL is a perpendicular bisector of EF, we get congruent segments FL=EL.

Answer: 4.25