Question

Answer the question

'The angles of a quadrilateral are in A.P. and the greatest angle is double the least. Find angles of the quadrilateral radian.'​

Answer 1

Answer :

$\bf The \ angles \ are \ \frac{\pi}{3} ,\frac{4\pi}{9} , \frac{5\pi}{9} , \frac{2\pi}{3}$

Step-by-step explanation :

$\underline{\underline{\bf Arithmetic \ Progression(AP):}}$

• It is the sequence of numbers such that the difference between any two successive numbers is constant.

Given,

⇒ The angles of a quadrilateral are in A.P.

Let the angles of the quadrilateral be

 a , a + d , a + 2d , a + 3d

The greatest angle = a + 3d

The least angle = a

We know,

 Sum of all the angles in the quadrilateral = 360°

      a + a + d + a + 2d + a + 3d = 360°

                   4a + 6d = 360°

               2(2a + 3d) = 2(180°)

                  2a + 3d = 180° --- [ 1 ]

It is also given,

 ⇒ the greatest angle is double the least.

        a + 3d = 2(a)

         a + 3d = 2a

         2a - a = 3d

              a = 3d

Substitute a = 3d in equation [1]

  2a + 3d = 180°

 2(3d) + 3d = 180°  

   6d + 3d = 180°

      9d = 180

        d = 180/9

         d = 20

→ a = 3d = 3(20) = 60°

The angles are :

• a = 60°
• a + d = 60 + 20 = 80°
• a + 2d = 60 + 2(20) = 100°
• a + 3d = 60 + 3(20) = 120°

Converting to radians :

    $\boxed{\bf 1^{\circ}=\frac{\pi}{180} \ rad }$

$\implies 60^{\circ} =60 \times \frac{\pi}{180} =\frac{\pi}{3} \ rad \\\\ \implies 80^{\circ}=80^{\circ} \times \frac{\pi}{180} = \frac{4\pi}{9} \ rad \\\\ \implies 100^{\circ} =100^{\circ} \times \frac{\pi}{180} =\frac{5\pi}{9} \ rad \\\\ \implies 120^{\circ} =120^{\circ} \times \frac{\pi}{180} =\frac{2\pi}{3} \ rad$