Question

One angle of a triangle is \[\dfrac{2}{3}x\] gradients and another is \[\dfrac{2}{3}x\] degrees while the third is \[\dfrac{\pi x}{75}\] radians. Express all the angles in degrees.

Answer 1

Hint: First of all we will convert all the given angles of a triangle in degree. We know that 1 gradient \[={{0.9}^{\circ }}={{\left( \dfrac{9}{10} \right)}^{\circ }}\] and 1 radian \[={{\dfrac{180}{\pi }}^{\circ }}\]. So by using these conversions, we will convert the given angle of the triangle from gradients and radians to degree. Then, we will use the property of a triangle that the sum of interior angles equals to \[{{180}^{\circ }}\].

Complete step-by-step answer:

We have been given the angles of a triangle as \[\dfrac{2}{3}x\] gradients, \[\dfrac{2}{3}x\] degrees and \[\dfrac{\pi x}{75}\] radians.
We know that 1 gradient \[={{0.9}^{\circ }}={{\left( \dfrac{9}{10} \right)}^{\circ }}\]
So, \[\dfrac{2}{3}x\] gradients \[={{\left( \dfrac{2x}{3}\times \dfrac{9}{10} \right)}^{\circ }}={{\left( \dfrac{18x}{30} \right)}^{\circ }}={{\left( \dfrac{3}{5}x \right)}^{\circ }}\]
Once again, we know that 1 radian \[={{\dfrac{180}{\pi }}^{\circ }}\]
So, \[\dfrac{\pi x}{75}\] radians \[={{\left( \dfrac{\pi x}{75}\times \dfrac{180}{\pi } \right)}^{\circ }}={{\left( \dfrac{180x}{75} \right)}^{\circ }}=\dfrac{12}{5}{{x}^{\circ }}\]
Now we have the angles of the triangle in degrees as follows:
\[{{\left( \dfrac{3}{5}x \right)}^{\circ }},{{\left( \dfrac{2}{3}x \right)}^{\circ }},{{\left( \dfrac{12}{5}x \right)}^{\circ }}\]
We know that the sum of interior angles of a triangle is equal to \[{{180}^{\circ }}\].
\[\Rightarrow \dfrac{3}{5}x+\dfrac{2}{3}x+\dfrac{12}{5}x={{180}^{\circ }}\]
On taking LCM of 5, 3 and 5, we get as follows:
\[\begin{align}
  & \Rightarrow \dfrac{3\left( 3x \right)+5\left( 2x \right)+3\left( 12x \right)}{15}={{180}^{\circ }} \\
 & \Rightarrow \dfrac{9x+10x+36x}{15}={{180}^{\circ }} \\
 & \Rightarrow \dfrac{55x}{15}={{180}^{\circ }} \\
\end{align}\]
On multiplying by 15 on both the sides, we get as follows:
\[\begin{align}
  & \Rightarrow 15\times \dfrac{55x}{15}={{180}^{\circ }}\times 15 \\
 & \Rightarrow 55x={{180}^{\circ }}\times 15 \\
\end{align}\]
On dividing the equation by 55, we get as follows:
\[\begin{align}
  & \Rightarrow \dfrac{55x}{55}=\dfrac{{{180}^{\circ }}\times 15}{55} \\
 & \Rightarrow x=\dfrac{{{540}^{\circ }}}{11} \\
 & \Rightarrow \dfrac{3}{5}x=\dfrac{3}{5}\times \dfrac{{{540}^{\circ }}}{11}=\dfrac{{{324}^{\circ }}}{11} \\
 & \Rightarrow \dfrac{2}{3}x=\dfrac{2}{3}\times \dfrac{{{540}^{\circ }}}{11}=\dfrac{{{360}^{\circ }}}{11} \\
 & \Rightarrow \dfrac{12}{5}x=\dfrac{12}{5}\times \dfrac{{{540}^{\circ }}}{11}=\dfrac{{{1296}^{\circ }}}{11} \\
\end{align}\]
Therefore, the angles of the triangle in degrees are \[\dfrac{{{324}^{\circ }}}{11},\dfrac{{{360}^{\circ }}}{11},\dfrac{{{1296}^{\circ }}}{11}\].

Note: Be careful while solving for ‘x’ and take the LCM of 5, 3 and 5 properly. Also remember that 1 gradient \[={{0.9}^{\circ }}={{\left( \dfrac{9}{10} \right)}^{\circ }}\] and 1 radian \[={{\dfrac{180}{\pi }}^{\circ }}\]. We also have to be careful while changing the radians into degrees and not get confused in the conversion that 1 radian \[={{\dfrac{180}{\pi }}^{\circ }}\]. In a hurry, we might take 1 radian \[={{\dfrac{\pi }{180}}^{\circ }}\] which is absolutely wrong.