Question

the five angles of a pentagon are in ap and the greatest angle is three times the least angle find the angles in degrees and radians

Answer 1

Answer:

54°,  81°,  108°,  135°,  162°.

3π/10,   9π/20,   3π/5,  3π/4,   9π/10

Step-by-step explanation:

Let a be the smallest angle and d the common difference.  So the angles are:

a, a+d, a+2d, a+3d, a+4d.

As the greatest angle is 3 times the least, we have:

a + 4d = 3a  =>  4d = 2a  =>  2d = a      ... (1)

The sum of the angles of an n sided polygon is (n-2)×180°.  So adding up our five angles gives

5a + 10d = 3 × 180°

=> a + 2d = 3 × 36° = 108°

=> 2a = 108  °                     [ used equation (1) ]

=> a = 54°

and d = a/2 = 27°

So the five angles are

54°,  81°,  108°,  135°,  162°.

To convert to radians, we multiply by π/180°.  So...

a = 54π / 180 = 3π / 10 = 6π / 20

and

d = a/2 = 3π / 20.

From here, the five angles are

6π / 20,   9π / 20,   12π / 20,   15π / 20,   18π / 20

which simplify to

3π / 10,   9π / 20,   3π / 5,  3π / 4,   9π / 10