Angles of triangle are in A.P. If the number of degrees in the smallest to the number of radians in the largest is as60:π, then the smallest angle isa)30∘b)20∘c)40∘d)none of these.Correct answer is option 'A'. Can you explain this answer?

Question

Angles of triangle are in A.P. If the number of degrees in the smallest to the number of radians in the largest is as60:&#960;, then the smallest angle isa)30&#8728;b)20&#8728;c)40&#8728;d)none of these.Correct answer is option 'A'. Can you explain this answer?

Answer

Let the angles of the triangle are (a - d), a, (a + d)
Now, in a triangle, the sum of all angles equal to 180 degree
⇒ (a - d) + a + (a + d) = 180
⇒ a - d + a + a + d = 180
⇒ 3a = 180
⇒ a = 180/3
⇒ a = 60
Now, the angle are (60 - d), 60, (60 + d)
Now, 60 - d is the least and 60 + d is the greatest angle.
Now, (60 + d)° = {(60 + d) * (π/180)}c
Given that, number of radians in the greatest angle/number of degrees in the least one = π/60
⇒ {(60 + d) * (π/180)}c /(60 - d) = π/60
⇒ (60 - d)/{(60 + d) * (π/180)}c = 60/π
⇒ 180(60 - d)/{(60 + d) * π} = 60/π
⇒ 180(60 - d)/(60 + d) = 60
⇒ (60 - d)/(60 + d) = 60/180
⇒ (60 - d)/(60 + d) = 1/3
⇒ 3(60 - d) = (60 + d)
⇒ 180 - 3d = 60 + d
⇒ 180 - 60 = 3d + d
⇒ 4d = 120
⇒ d = 120/4
⇒ d = 30

Answer

Answer

Given information:
- Angles of the triangle are in arithmetic progression (A.P.).
- The number of degrees in the smallest angle to the number of radians in the largest angle is in the ratio 60:1.

To find:
The measure of the smallest angle of the triangle.

Solution:
Let the three angles of the triangle be A, A + d, and A + 2d, where A is the measure of the smallest angle and d is the common difference.

Converting degrees to radians:
We know that π radians = 180 degrees.
Therefore, 1 radian = (180/π) degrees.
Let the measure of the largest angle in radians be L. So, L = (A + 2d) * (180/π).

Given ratio:
According to the given information, the ratio of the smallest angle in degrees to the largest angle in radians is 60:1.
So, A : L = 60 : 1.

Converting the ratio to radians:
Since 1 radian = (180/π) degrees, we can write the ratio in radians as:
A * (180/π) : L = 60 : 1.

Substituting the values:
Substituting the values of L and A * (180/π), we get:
(A * (180/π)) / ((A + 2d) * (180/π)) = 60 / 1.

Cancelling out the common terms:
Cancelling out the common terms, we are left with:
1 / (A + 2d) = 60.

Cross-multiplication:
Cross-multiplying, we get:
60(A + 2d) = 1.

Simplifying the equation:
Simplifying the equation, we get:
60A + 120d = 1.

Substituting values for d:
Since d is the common difference, it must be a multiple of 1/60. Let's assume d = k/60, where k is a positive integer.

Substituting the value of d in the equation:
Substituting the value of d in the equation, we get:
60A + 120(k/60) = 1,
60A + 2k = 1.

Checking the options:
Now, let's check the given options:
a) 30: If we substitute A = 30, the equation becomes 60(30) + 2k = 1, which is not possible since the left-hand side is even.
b) 20: If we substitute A = 20, the equation becomes 60(20) + 2k = 1, which is not possible since the left-hand side is even.
c) 40: If we substitute A = 40, the equation becomes 60(40) + 2k = 1, which is not possible since the left-hand side is even.
d) None of these.

Correct answer:
None of the given options is the correct answer.

Therefore, there seems to be an error in the given question or the answer options, as the equation does not have a solution for

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