Question

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Answer 1

Answer:

n = 3

Step-by-step explanation:

Given, Smallest angle = a = 52°

Common difference = d = 8°.

Let the polygon has 'n' number of sides.

Given that the angles are in A.P.

Let the angles of the polygon are:

a,a + d, a + 2d...

(i) Sum of n terms:

= (n/2)[2a + (n - 1) * d]

= (n/2)[104 + (n - 1) * 8]

= (n/2)[104 + 8n - 8]

= (n/2)[8n + 96]

= 4n[n + 12]

(ii)

Sum of all the angles of polygon with n sides = (n - 2) * 180

From (i) & (ii), we get

⇒ 4n(n + 12) = (n - 2) * 180

⇒ 4n² + 48n = 180n - 360

⇒ 4n² - 132n + 360 = 0

⇒ n² - 33n + 90 = 0

⇒ n² - 30n - 3n + 90 = 0

⇒ n(n - 30) - 3(n - 30) = 0

⇒ (n - 3)(n - 30) = 0

⇒ n = 3, 30{a + 29d = 52 + 29 * 8 = 284 > 180°.. Not possible }

⇒ n = 3.

Therefore, the value of n is 3.

Hope it helps!

Answer 2

Question

The interior angles of a polygon are in AP.The smallest angle is 52° and the common difference is 8°.Find number it sides in polygon.

ANSWER

Given,

Smallest angle(a) = 52°

Common difference(d) = 8°

Hence the AP formed will be =>

52,60,68,76,84.........

We know,

Sum of interior angles of a polygon=(n-2)180°

Hence,the equation formed is=>

n/2{2a + (n-1)d} = (n-2)180°

=>n/2{2×52 + (n-1)8} = (n-2)180°

=>n/2(104 + 8n -8) = 180n - 360°

=>n/2(96 +8n) = 180n - 360°

=>48n+ 4n^2 = 180n - 360°

=>4n^2 - 132n + 360 = 0

=>4(n^2 - 33n + 90) =0

So,on dividing both sides by 4 we get =>

n^2 - 33n + 90 =0

Since this is a quadratic equation we will have two vales for n.

=>n^2 -30n -3n +90 =0

=>n(n-30) -3(n-30)=0

=>(n-30)(n-3) =0

Hence, either n= 3 or

n = 30(neglected).

So, we get that the number of sides of polygon is 3

thanks