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

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Last updated date: 24th Jun 2024
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Answer
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Hint: Let us assume the angles $(x+y){}^\circ ,x{}^\circ $ and $(x-y){}^\circ $ which are in A.P. Now we know that since these are angles of triangles, the sum of angles will be $180{}^\circ $. After that, apply the condition given and simplify it.

Complete step-by-step answer:
We are given that the angles of a triangle are in A.P.
If $y$ is a common difference, let us consider the angles $(x+y){}^\circ ,x{}^\circ $ and $(x-y){}^\circ $.
Now we know that since these are angles of triangles, the sum of angles will be $180{}^\circ $.
$\Rightarrow$ $x-y+x+x+y=180{}^\circ $
Simplifying we get,
$\Rightarrow$ $3x=180{}^\circ $
Now divide above equation by $3$ we get,
$\Rightarrow$ $x=60{}^\circ $
Also, we are given that, the number of degrees in the least is to the number of radians in the greatest as $60:\pi $, we get,
$\Rightarrow$ $\dfrac{(x-y)}{(x+y)\dfrac{\pi }{180}}=\dfrac{60}{\pi }$
Simplifying we get,
$\Rightarrow$ $\dfrac{(x-y)}{(x+y)}=\dfrac{60}{\pi }\times \dfrac{\pi }{180}$
$\Rightarrow$ $\dfrac{(x-y)}{(x+y)}=\dfrac{1}{3}$
Now cross multiplying we get,
$\Rightarrow$ $3x-3y=x+y$
Again, simplifying we get,
$\Rightarrow$ $x-2y=0$ ……… (2)
Now substituting $x=60{}^\circ $ in equation (2).
$\Rightarrow$ $60-2y=0$
Simplifying we get,
$\Rightarrow$ $2y=60$
Now divide above equation by $2$ we get,
$\Rightarrow$ $y=30{}^\circ $
Now taking the three angles,
$\Rightarrow$ $(x+y){}^\circ =(60+30){}^\circ =90{}^\circ $
$\Rightarrow$ $x{}^\circ =60{}^\circ $
$\Rightarrow$ $(x-y){}^\circ =(60-30){}^\circ =30{}^\circ $
Therefore, we get the three angles as $30{}^\circ ,60{}^\circ $ and $90{}^\circ $.

Additional information:
Arithmetic Mean is simply the mean or average for a set of data or a collection of numbers. In mathematics, we deal with different types of means such as arithmetic mean, arithmetic harmonic mean, geometric mean and geometric harmonic mean. The term Arithmetic Mean is just used to differentiate it from the other “means” such as harmonic and geometric mean. The arithmetic mean is a good average. It is sometimes known as average. But, it cannot be used in some cases like, the distribution has open end classes, the distribution is highly skewed, averages are taken for ratios and percentages.

Note: Here we have considered the angles $(x+y){}^\circ ,x{}^\circ $ and $(x-y){}^\circ $. You can assume different angles, it is not necessary that you should assume different variables. Arithmetic mean is the simplest measure of central tendency and is the ratio of the sum of the items to the number of items.