The angles of a triangle are in the ratio 8 : 3 : 1. What is the ratio

1.	The angles of a triangle are in the ratio 8 : 3 : 1. What is the ratio of the longest side of the triangle to the next longest side?
Answer» The angles are \(\frac{8}{12}\times180°,\frac{3}{12}\times180°\) i.e., 120º, 45º and 15º. Also we know that the longest side is opposite the greatest angle and so on. ∴ Let the longest side opposite the greatest angle 120º be \(x\) and let the next longest side opposite angle 45º be y. Then, by the sine rule \(\frac{sin\,120°}{x}\) = \(\frac{sin\,45°}{y}\) ⇒ \(\frac{x}{y}=\frac{sin\,120°}{sin\,45°}=\frac{\frac{\sqrt3}{2}}{\frac{1}{\sqrt2}}=\frac{\frac{\sqrt3}{2}}{\frac{1}{\sqrt2}}=\frac{\sqrt6}{2}.\)

The angles of a triangle are in the ratio 8 : 3 : 1. What is the ratio of the longest side of the triangle to the next longest side?

Answer»

The angles are \(\frac{8}{12}\times180°,\frac{3}{12}\times180°\) i.e., 120º, 45º and 15º.

Also we know that the longest side is opposite the greatest angle and so on.

∴ Let the longest side opposite the greatest angle 120º be \(x\) and let the next longest side opposite angle 45º be y. Then, by the sine rule

\(\frac{sin\,120°}{x}\) = \(\frac{sin\,45°}{y}\)

⇒ \(\frac{x}{y}=\frac{sin\,120°}{sin\,45°}=\frac{\frac{\sqrt3}{2}}{\frac{1}{\sqrt2}}=\frac{\frac{\sqrt3}{2}}{\frac{1}{\sqrt2}}=\frac{\sqrt6}{2}.\)