Question

One angle of a triangle $\frac{2}{3}$x grades and another is $\frac{3}{2}$x degrees while the third is $\frac{\mathrm{\pi x}}{75}$ radians. Express all the angles in degrees.

Answer 1

One angle of the triangle = $\frac{2}{3}x\mathrm{grad}$
$$\begin{gathered} ={\left(\frac{2}{3}x\times \frac{9}{10}\right)}^{°}\left[\because 1\mathrm{grad}={\left({\displaystyle \frac{9}{10}}\right)}^{°}\right] \\ ={\left(\frac{3}{5}x\right)}^{°} \end{gathered}$$

Another angle = ${\left(\frac{3}{2}x\right)}^{°}$

$\because 1\mathrm{radian}={\left({\displaystyle \frac{180}{\mathrm{\pi }}}\right)}^{°}$
$$\begin{gathered} \mathrm{Third}\mathrm{angle}\mathrm{of}\mathrm{the}\mathrm{triangle}=\frac{x\mathrm{\pi }}{75}\mathrm{rad} \\ ={\left({\displaystyle \frac{180}{\mathrm{\pi }}}\times {\displaystyle \frac{x\mathrm{\pi }}{75}}\right)}^{°} \\ ={\left(\frac{12}{5}x\right)}^{°} \end{gathered}$$
Now,
$$\begin{gathered} \frac{3}{5}x+\frac{3}{2}x+\frac{12}{5}x=180(\mathrm{Angle}\mathrm{sum}\mathrm{property}) \\ \Rightarrow \frac{6\mathrm{x}+15\mathrm{x}+24\mathrm{x}}{10}=180 \\ \Rightarrow \frac{45\mathrm{x}}{10}=180 \\ \Rightarrow \mathrm{x}=40 \end{gathered}$$


Thus, the angles are:
$$\begin{gathered} {\left(\frac{3}{5}x\right)}^{°}=24° \\ \left(\frac{3}{2}x\right){}^{°}=60° \\ {\left(\frac{12x}{5}\right)}^{°}=96° \end{gathered}$$.