Question

the angle of the triangle are in AP if the smallest angle is one third the largest angle find the angle of the triangle​

Answer 1

Question:

The angles of a triangle are in AP. If the smallest angle is one third of the largest angle , find the angles of the triangle.

Answer:

30° , 60° , 90°

The smallest angle is 30°

The largest angle is 90°

Note:

• A sequence in which, the difference between the consecutive terms are same is called AP (Arithmetic Progression).

• Any AP is given as ; a , (a + d) , (a + 2d) , .....

• The nth term of an AP is given by ;

T(n) = a + (n - 1)d , where a is the first term and d is the common difference of the AP .

• Angle-sum-property of a triangle: The sum of all the three interior angles of a triangle is equal to 180° .

Solution:

It is given that ;

The angles of a triangle are in AP .

Thus,

Let the 1st , 2nd and 3rd angles be a , (a + d) and (a + 2d) .

Now,

We know that,

The sum of all the three interior angles of a triangle is 180° .

Thus;

=> a + (a + d) + (a + 2d) = 180°

=> 3a + 3d = 180°

=> 3(a + d) = 180°

=> a + d = 180°/3

=> a + d = 60° --------(1)

Also,

It is given that;

The smallest angle is 1/3rd of the largest angle.

Thus;

=> a = 1/3 of (a + 2d)

=> a = (1/3)(a + 2d)

=> a = (a + 2d)/3

=> 3a = a + 2d

=> 3a - a = 2d

=> 2a = 2d

=> a = d ---------(2)

Now,

Putting a = d , in eq-(1) , we get ;

=> a + d = 60°

=> d + d = 60°

=> 2d = 60°

=> d = 60°/2

=> d = 30°

Now,

Putting d = 30° , in eq-(2) , we get ;

=> a = d

=> a = 30°

Thus,

1st angle = a = 30°

2nd angle = a + d = 30° + 30° = 60°

3re angle = a + 2d = 30° + 2•30° = 30° + 60° = 90°

Hence,

The required angles of the given triangle are ;

30° , 60° , 90° .

Smallest angle = 30°

Largest angle = 90°

Answer 2

Answer:

$\huge{\underline{\underline{\mathfrak{Question\::}}}}$

The angles of a triangle are in AP. If the smallest angle is one third of the largest angle.Find the angles of the triangle.

$\textbf{\underline{Solution\::}}$

An AP is a term in which the difference between the consecutive terms is same.

_______________________

Let the angles of the triangle be as follows: x, (x+a) , (x + 2a)

Given that, smallest angle = 1/3 × largest angle.

=> x = 1/3 × (x+2a)

=> x = (x+2a)/3

=> 3x = x + 2a

=> 3x - x = 2a

=> 2x = 2a

=> x = a .............(eq.1)

________________________

Now, we know that sum of the angles of a triangle = 180°

According to question now:-

x + (x + a) + (x + 2a) = 180°

=> x + x + a + x + 2a = 180°

=> 3x + 3a = 180°

=> 3(x+a) = 180°

=> x + a = 180°/3

=> x + a = 60°

Putting the value of (eq.1) here,

=> x + x = 60°

=> 2x = 60°

=> x = 60°/2

=> x = 30°

______________________

Smallest angle = x = 30°

Largest angle=(x + 2a)= 30°×3 =90°

As the angles are in AP,

Middle angle = (30 + (90-30)/2)°

=> Middle angle = (30+30)°

=> Middle angle = 60°

_________________________

The angles of the triangle are : 30° , 60° and 90°.

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