Question

(x) The largest angle of a triangle is twice the sum of the other two. The smallest is one-fifth of the largest. Determine all the angles in degrees.

Answer 1

$$\begin{gathered} \mathrm{Let}\mathrm{the}\mathrm{smallest}\mathrm{angle}\mathrm{of}\mathrm{the}\mathrm{triangle}=x° \\ \mathrm{Other}\mathrm{angle}=y° \\ \therefore \mathrm{Largest}\mathrm{angle}=\left[2\left(x+y\right)\right]°...\left(1\right) \\ \mathrm{It}\mathrm{is}\mathrm{given}\mathrm{that}\text{the}\mathrm{smallest}\mathrm{angle}\mathrm{is}\mathrm{one}\mathrm{fifth}\mathrm{of}\mathrm{the}\mathrm{largest}.\text{So,} \\ x=\frac{1}{5}\times 2\left(x+y\right) \\ \Rightarrow 5x=2x+2y \\ \Rightarrow 3x=2y...\left(2\right) \\ \mathrm{As}\text{the}\mathrm{sum}\mathrm{of}\mathrm{all}\mathrm{the}\mathrm{angles}\mathrm{of}a\mathrm{triangle}\mathrm{is}180°, \\ x+y+2\left(x+y\right)=180 \\ \Rightarrow 3x+3y=180 \\ \Rightarrow 2y+3y=180\left\{\mathrm{From}\left(2\right)\right\} \\ \Rightarrow 5y=180 \\ \therefore y=36 \\ \mathrm{Substituting}y=36\mathrm{in}\mathrm{eq}\left(2\right)\mathrm{we}\mathrm{get}: \\ 3x=2\times 36 \\ \therefore x=24 \\ \mathrm{Hence}, \\ \text{s}\mathrm{mallest}\mathrm{angle}=24° \\ \text{o}\mathrm{ther}\mathrm{angle}=36° \\ \text{l}\mathrm{argest}\mathrm{angle}=2\left(24+36\right)=120° \end{gathered}$$