Question

# The angles of a triangle are in A.P. The number of degrees in the least is to be the number of radians in the greatest as $60:\pi .$ Then the greatest angle is$A.\;\;\;\;\;120^\circ \\ B.\;\;\;\;\;90^\circ \\ C.\;\;\;\;\;135^\circ \\ D.\;\;\;\;\;105^\circ \\$

Hint:Using the knowledge of the properties of a triangle and A.P. we will approach the solution to our problem. Using properties like
1) the sum of all angles of a triangle is $180^\circ$
2) common difference of consecutive terms of an A.P. remains constant.
3) ${1^o} = \dfrac{\pi }{{180}}radians$
we will form equations to solve for the required quantity.

Given data: The angles of a triangle are in A.P.
$\dfrac{{least{\text{ }}angle(in{\text{ }}\deg rees)}}{{greatest{\text{ }}angle(in{\text{ }}radians)}} = \dfrac{{60}}{\pi }$
Now, let us assume that the angles be x, y and z where $\left( {x < y < z} \right)$
From the above assumption, we can say that,
‘x’ is the least angle and ‘z’ is the greatest angle
Since x, y and z are in A.P., the common difference remain constant i.e.,
$y - x = z - y \\ \Rightarrow 2y = x + z....................(i) \\$
It is also given that,
$\dfrac{{least{\text{ }}angle(in{\text{ }}\deg rees)}}{{greatest{\text{ }}angle(in{\text{ }}radians)}} = \dfrac{{60}}{\pi }$
On using the fact that ${1^o} = \dfrac{\pi }{{180}}radians$, we get,
$\dfrac{x}{{z(\dfrac{\pi }{{180}})}} = \dfrac{{60}}{\pi }$
$\Rightarrow x = z(\dfrac{\pi }{{180}})\dfrac{{60}}{\pi } \\ \Rightarrow x = \dfrac{z}{3}...........................(ii) \\$
We also that sum of all angles of a triangle is $180^\circ$ i.e.,
$x + y + z = {180^o}.................(iii)$
From equation (i) and (ii), we get,
$2y = \dfrac{z}{3} + z \\ \Rightarrow 2y = \dfrac{{4z}}{3} \\ \Rightarrow y = \dfrac{{2z}}{3} \\$
Now putting the value of ‘x’ and ‘y’ in equation(iii), we get,
$\Rightarrow \dfrac{z}{3} + \dfrac{{2z}}{3} + z = {180^o} \\ \Rightarrow \dfrac{{z + 2z + 3z}}{3} = {180^o} \\ On{\text{ }}Further{\text{ }}simplification{\text{ }}we{\text{ }}get, \\ \Rightarrow \dfrac{{6z}}{3} = {180^o} \\ \Rightarrow 2z = {180^o} \\ \Rightarrow z = {90^o} \\$
From the assumption we made it is clear that ‘z’ is the greatest angle of the triangle i.e. ${90^o}$
Hence, the correct option is (B).

Note: While writing the ratio given as per the question do not forget to convert the greatest angle in radians as if not done the answer will not match the correct option
Additional information: In any triangle, an angle comes out to be right angle then it will be the greatest angle as no other angle comes out to be greater than 90° , as the sum of angles will exceed $180^\circ$which is not possible.