The words roo, roux, rue sound the same but have different meanings and spellings. Why do roo, roux, rue sound the same even though they are completely different words?
The answer is simple: roo, roux, rue are homophones of the English language.
Peace; quietness.
Rest; stillness.
Short form of kangaroo.
Quintana Roo, a state of Mexico.
A mixture of flour and fat cooked together and used as a thickening.
To feel regret, remorse, or sorrow for.
To feel regret, remorse, or sorrow.
Sorrow; regret: "To their rue, the Social Democrats have to acknowledge that the Conservative-Liberal coalition has captured the center where elections are won” ( Elizabeth Pond).
Any of various aromatic southwest Asian or Mediterranean plants of the genus Ruta, especially the ornamental R. graveolens, having bipinnately compound leaves that yield an acrid volatile oil formerly used in medicine.
Definitions from Wiktionary, Creative Commons Attribution/Share-Alike License, from The American Heritage® Dictionary of the English Language, 4th Edition and Wordnik.
Homophones (literally "same sound") are usually defined as words that share the same pronunciation, regardless of how they are spelled.
If they are spelled the same then they are also homographs (and homonyms); if they are spelled differently then they are also heterographs (literally "different writing").