Euthanasia is known by different terms such as mercy killing, assisted suicide, etc. It is the act of assisting someone in his or her own death who is terminally ill, suffering, and in great pain. The goal of the assisted suicide is to prevent the continuation of pain.
The Bible tells us that we are not to murder (Exodus 20:13). Murder is the unlawful taking of life, and killing is the lawful taking of life. Technically speaking, if a nation said that euthanasia was legal, then on a human level it would not be murder. But as societies often legislate moral issues in contradiction to the Bible and just because a society might say that euthanasia is good, does not mean that it is. We are to obey God rather than men (Acts 5:29).
We are made in the image of God (Gen. 1:26), and it is the Lord God who gives us life (Job 33:4) and who has numbered our days (Job 14:5). This means that God is the sovereign Lord who determines the day that we die. Therefore, we are not to usurp God's authority