Question

The angles of a triangle are in AP and the least angle is ${{30}^{\circ }}$ . The greatest angle in radians is:

a)$\dfrac{7\pi }{12}$

b) $\dfrac{2\pi }{3}$

c)$\dfrac{5\pi }{6}$

d)$\dfrac{\pi }{2}$

Answer 1

Hint: An arithmetic progression is a sequence of numbers with the same successive difference. Suppose the angles of the triangle in the form of an A.P. Now use the fundamental rule of a triangle that the sum of interior angles of a triangle is always ${{180}^{\circ }}$ . Use $\pi $ radian$={{180}^{\circ }}$ whenever required.

Complete step-by-step answer:
We know that A.P stands for arithmetic progression and is defined as a sequence with the same successive difference of two continuous numbers or terms. It means general sequence of an A.P can be given as
a, a + d, a + 2d, a + 3d……………………..
Where a is the first term and the common difference that we can observe that difference between two consecutive terms are the same i.e. ‘d’.
Now, coming to the question, it is given that angles of the given triangle are in A.P and need to determine the greatest angle if the smallest angle is given as ${{30}^{\circ }}$ . So, suppose angles of the triangle are a, a + d, a + 2d where d is positive. It means the relation between angles can be given as
$a\le a+d\le a+2d$. So, a is the smallest angle of the triangle and (a + 2d) is the largest angle of the given triangle. Hence, we have the smallest angle i.e. ‘a’ is 30 we get,
a = 30……………….(i)
So, the A.P can be given as
30, 30 + d, 30 + 2d……………..(ii)
Now, we know the fundamental property of a triangle is that the sum of interior angles of a triangle will be ${{180}^{\circ }}$ always. Hence, the sum of the equation (ii) is 180. So, we get
30 + 30 + d + 30 + 2d =180.
90 + 3d = 180
3d = 90
$d=\dfrac{90}{3}=30$
Hence, angles of the triangle can be given as
$30,30+30,30+30\times 2$ or
30, 60, 30 + 60 or 30, 60, 90.
Hence, the greatest angle of the given triangle is ${{90}^{\circ }}$ . As, we can observe that angles in the options are in radians, so we need to convert ${{90}^{\circ }}$ to radian. So, we know that ${{180}^{\circ }}$ can be written as $\pi $ radian. So, ${{1}^{\circ }}$ can be written as $\dfrac{\pi }{180}$ radian. Hence, ${{90}^{\circ }}$ can be given as $\dfrac{\pi }{180}\times 90$ radian = $\dfrac{\pi }{2}$ radian.
So, option (d) is the correct answer.

Note: One may suppose the three terms of an A.P as ( a- d), a,(a + d) where the common difference is ‘d’ with the first term ‘a’. It simplifies the relation when we add these terms and equate the sum to ${{180}^{\circ }}$ , as – d and d will cancel out to each other. Hence, one may use these terms as well for the angles of the given triangle.
One may suppose ‘a’ as the greatest angle and (a + 2d) as smallest, but d will be negative this time. So, don’t confuse the terms ‘greatest’ and ‘smallest’ angle in the problem. Be clear and careful with it as well.