will sapphire amd ati r7 250 1gb ddr5 work on asus h-61m-k mother bord

No problems at all, it will work. There is basically only 1 thing which determines the compatibility of motherboard and graphics card: does your mobo have the appropriate slot (PCI-E x16)? It does, so check.

Two more things additionally determine the ability to actually USE the card:

1) Do you have enough space in your case? --> this card is fairly small and fits most cases, so check
2) Is your PSU strong enough to feed the card properly? --> this card uses up to 65W only, and does not even require external power cables, so you are all good provided you have even half-decent PSU above 400W.