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The angle of a triangle are in A.P. If the greatest angle is twice the smallest angle, find the angles of the triangles using matrices.



Answer :

Let's denote the angles of the triangle as a, b, and c, with a < b < c. Given that the angles are in Arithmetic Progression (A.P.), we can express the angles as: a, a + d, and a + 2d, where d is the common difference.

Since the sum of the angles in any triangle is always 180 degrees, we have:

  • a + (a + d) + (a + 2d) = 180

Simplifying this, we get:

  • 3a + 3d = 180
  • a + d = 60

Additionally, it's given that the greatest angle is twice the smallest angle. Therefore, we have:

  • a + 2d = 2a

Solving for d, we get:

  • d = a

Substituting d = a into a + d = 60, we find:

  • a + a = 60
  • 2a = 60
  • a = 30 degrees

Consequently, the other angles are:

  • a + d = 30 + 30 = 60 degrees
  • a + 2d = 30 + 60 = 90 degrees

So the angles of the triangle are 30 degrees, 60 degrees, and 90 degrees. While matrices are typically used for more complex systems of equations, solving this problem relies on algebraic manipulation of the given arithmetic properties.

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