Question

Please tell the name of the polygon whose smallest angle is 120° and whose difference between any two consecutive interior angle is 5°

If possible you can draw that polygon and name it?

But don't answer if you don't know
^_^​

Answer 1

Answer:

$\large\boxed{\sf{9\;\;sides\;\:or\:\;16\;\;sides}}$

Step-by-step explanation:

So, it's given that :

Smallest angle of polygon = 120

Also, there is difference of 5° between consecutive interior angles.

Therefore, angles will be in the order :

120°, 125° , 130°,........

Clearly it's an AP

where,

• First term, a = 120
• Common difference, d = 5

Now, we know that sum of all interior angles of a polygon having 'n' sides is given by $\red{(n-2)180}$

Therefore, the sum of all angles will be equal to the above formula.

Also, Sum of 'n' terms of an AP is given by $\red{\dfrac{n}{2} \left[2a +( n - 1)d\right]}$

So, both of the formulae must be same.

Thus, substituting the values, we get,

$= > \dfrac{n}{2} (2 \times 120 + (n - 1)5) = (n - 2)180 \\ \\ = > \frac{n}{2} (240 + 5n - 5) = (n - 2)180 \\ \\ = >5 {n}^{2} + 235n = 360n - 720 \\ \\ = > 5 {n}^{2} -1 25n + 720 = 0 \\ \\ = > {n}^{2} -2 5n + 144 = 0 \\ \\ = > {n}^{2} - 16n - 9n + 144 = 0 \\ \\ = > n(n - 16) - 9(n - 16) = 0 \\ \\ = > (n - 16)(n - 9) = 0 \\ \\ = > \sf{ n = 9 \: \: \: or \: \: \: n =16}$

Hence , The polygon can be of 9 sides or 16 sides.

Answer 2

||✪✪ CORRECT QUESTION ✪✪||

The interior angles of regular polygon are in A.P. The smallest angle is 120 degree and the common difference is 5 degree. Find the name of the polygon. ?

|| ★★ FORMULA USED ★★ ||

→ Sum of interior angles of regular polygon is = (n-2) * 180° [ where n is number of sides of Regular Polygon ] .

→ Sum of n terms of An AP = (n/2) [ 2a + (n-1)d] where a os first term of AP and d is common Difference .

→ A regular Polygon with 9 Equal sides is called as Nonagon .

→ A regular Polygon with 16 Equal sides is called as Hexadecagon.

|| ✰✰ ANSWER ✰✰ ||

we have given that , Smallest angle is 120° and common Difference is 5° ,,

Let us assume that , The regular Polygon has n Equal sides ....

So, comparing their Sum of interior angles with both the formula ( Told Above) , we get,

→ (n/2) [ 2a + (n-1)d] = (n - 2)*180°

→ (n/2) [ 2 * 120 + (n-1) * 5 ] = 180n - 360°

→ (n/2) [ 240 + 5n - 5 ] = 180n - 360°

→ n ( 235 + 5n ) = 360n - 720°

→ 235n + 5n² = 360n - 720°

→ 5n² + 235n - 360n + 720° = 0

→ 5n² - 125n + 720 = 0

→ 5(n² - 25n + 144) = 0

→ n² - 25n + 144 = 0

Splitting the Middle Term now,

→ n² - 16n - 9n + 144 = 0

→ n(n - 16) - 9(n - 16) = 0

→ (n - 16)(n - 9) = 0

Putting both Equal to 0 now,

→ n - 16 = 0

→ n = 16 .

or,

→ n - 9 = 0

→ n = 9.

So, as told above,

→ Regular regular Polygon with 9 Equal sides is called as Nonagon .

And,

→ Regular Polygon with 16 Equal sides is called as Hexadecagon.