Question

One angle of a triangle is $\dfrac{2x}{3}$ grad, another is $\dfrac{3x}{2}$ degrees, while the third is $\dfrac{\pi x}{75}$ radians. Express all angles in degrees,
A.Hence three angles of the triangle are $43{}^\circ ,30{}^\circ ,30{}^\circ $
B.Hence three angles of the triangle are $24{}^\circ ,60{}^\circ ,96{}^\circ .$
C.Hence three angles of the triangle are $74{}^\circ ,27{}^\circ ,98{}^\circ .$
D.Hence three angles of the triangle are $30{}^\circ ,60{}^\circ ,90{}^\circ .$

Answer 1

Hint:Convert all the units in degrees first so that there will be no confusion. Then use the concept given by “The sum of three angles of a triangle is always \[180{}^\circ \]” , you will get the value of ‘x’. Then put the value of ‘x’ in all angles to get the final answer.

Complete step by step answer:
As we have given the angles of a triangle therefore we will write the given angles with using some notations therefore the given data can be written as,
$\angle A=\dfrac{2x}{3}grad$, \[\angle B=\dfrac{3x}{2}degrees\], And \[\angle C=\dfrac{\pi x}{75}radians\]
As we have asked to find all the angles in degrees therefore we will convert all the angles in degrees as follows,
Consider,
$\angle A=\dfrac{2x}{3}grad$
To convert the angle from grad to degrees we have to multiply the angle by \[\dfrac{180}{200}\], therefore by multiplying \[\angle A\] by \[\dfrac{180}{200}\] in the above equation we will get,
\[\therefore \angle A=\left( \dfrac{2x}{3}\times \dfrac{180}{200} \right)Degrees\]
\[\therefore \angle A=\dfrac{6x}{10}Degrees,\]
\[\therefore \angle A=\dfrac{3x}{5}Degrees\] …………………………………………………………………… (1)
As the \[\angle B\] is already in degrees therefore there is no need of conversion, therefore we will get,
\[\angle B=\dfrac{3x}{2}Degrees\] ……………………………………………………………………... (2)
Also consider,
\[\angle C=\dfrac{\pi x}{75}radians\]
To convert radians to degrees we have to multiply the above angle by \[\dfrac{180}{\pi }\]. Therefore by multiplying the above equation by \[\dfrac{180}{\pi }\], we will get,
\[\angle C=\left( \dfrac{\pi x}{75}\times \dfrac{180}{\pi } \right)Degrees\]
\[\therefore \angle C=\left( \dfrac{x}{75}\times 180 \right)Degrees\]
\[\therefore \angle C=\dfrac{12x}{5}Degrees\] ……………………………………………………………….. (3)
As we have converted all the angles in degrees therefore by using the concept given below we have to find the value of ‘x’ so that we can find the angles,
Concept:
The sum of three angles of a triangle is always \[180{}^\circ \]
As \[\angle A\], \[\angle B\] and \[\angle C\] are the angles of a triangle therefore according to concept given above the sum of these angles will be \[180{}^\circ \] therefore we can write,
\[\therefore \angle A+\angle B+\angle C=180{}^\circ \]
If we substitute the values of equation (1), equation (2) and equation (3) in the above equation we will get,
\[\therefore \left( \dfrac{3x}{5} \right)Degrees+\left( \dfrac{3x}{2} \right)Degrees+\left( \dfrac{12x}{5} \right)Degrees=180Degrees\]
\[\therefore \dfrac{3x}{5}+\dfrac{3x}{2}+\dfrac{12x}{5}=180\]
\[\therefore \dfrac{3x}{2}+\dfrac{3x}{5}+\dfrac{12x}{5}=180\]
\[\therefore \dfrac{3x}{2}+\dfrac{15x}{5}=180\]
\[\therefore \dfrac{15x+30x}{10}=180\]
\[\therefore \dfrac{45x}{10}=180\]
\[\therefore x=180\times \dfrac{10}{45}\]
\[\therefore x=20\times \dfrac{10}{5}\]
\[\therefore x=4\times 10\]
\[\therefore x=40{}^\circ \] ……………………………………………………………….. (4)
Now if we put the value of equation (4) in equation (1) we will get,
\[\therefore \angle A=\dfrac{3\times 40}{5}Degrees\]
\[\therefore \angle A=\left( 3\times 8 \right){}^\circ \]
\[\therefore \angle A=24{}^\circ \] ………………………………………………………….. (5)
Also we will put the value of equation (4) in equation (2) therefore we will get,
\[\angle B=\dfrac{3\times 40}{2}Degrees\]
\[\therefore \angle B=\left( 3\times 20 \right){}^\circ \]
\[\therefore \angle B=60{}^\circ \] …………………………………………………………. (6)
Likewise we will put the value of equation (4) in equation (3), therefore we will get,
\[\therefore \angle C=\dfrac{12\times 40}{5}Degrees\]
\[\therefore \angle C=\left( 12\times 8 \right){}^\circ \]
\[\therefore \angle C=96{}^\circ \] ……………………………………………………………. (7)
From equation (5), equation (6) and equation (7) we can write the final answer as,
The three angles of a triangle in degrees are \[24{}^\circ \], \[60{}^\circ \], \[96{}^\circ \].
Therefore the correct answer is option (b).

Note: Do remember to convert all angles in one unit before adding them (prefer in degrees as it reduces the calculations) otherwise you will definitely get a wrong answer.