Question

Find the length of side x in simplest radical form with a rational denominator of a triangle that one side is 9 the other is x and there is 2 degrees that are 60 degrees and 20 degrees

Answer 1

I’m missing which side corresponds to which angle. Could you clarify or upload the diagram? For example, do you mean:

- A) The triangle’s angles are 60°, 20°, 100°, and the side of length 9 is opposite the 20° angle and x is opposite the 60° angle?
- B) Same angles but 9 is opposite 60° and x is opposite 20°?
- C) The side of length 9 and the side of length x meet at the 60° angle (i.e. 60° is the included angle)?

If you meant case A (9 opposite 20° and x opposite 60°), then by the Law of Sines
x / sin 60° = 9 / sin 20°,
so
x = 9 · sin 60° / sin 20° = (9·(√3/2)) / sin 20° = (9√3) / (2 sin 20°).

If you meant case B (9 opposite 60° and x opposite 20°),
x = 9 · sin 20° / sin 60° = 9·sin 20° / (√3/2) = 6√3 · sin 20°.

Note: sin 20° does not simplify to a simple expression in square roots (it requires higher-degree radicals), so (9√3)/(2 sin 20°) is the simplest “nice” exact form unless you want the complicated radical expression involving cube roots. If you tell me which labeling is correct I’ll finish the answer (or give a decimal value).