Question

The interior angles of a convex polygon are in A.P. If the smallest angle is \[{{100}^{\circ }}\] and the common difference is \[{{4}^{\circ }}\], then find the number of sides. Choose the correct option
A. 5
B. 7
C. 36
D. 44

Answer 1

Hint: Use the property that for convex polygons the sum of all interior angles as \[(n-2){{180}^{\circ }}\]. Also, use the sum of n terms of the AP which is given by the formula \[{{S}_{n}}=\dfrac{n}{2}\left[ 2a+(n-1)d \right]\].

Complete step by step answer:
In the question, we have to find the number of sides of the convex polygon if interior angles of a convex polygon are in A.P. and the smallest angle is \[{{100}^{\circ }}\] and the common difference is \[{{4}^{\circ }}\].
For the convex polygon the, any interior angle is always less than \[{{180}^{\circ }}\].
So, the first interior angle is \[{{100}^{\circ }}\], next the common difference is \[{{4}^{\circ }}\]. Now, let there are n sides so there will be n angles in A.P. that will be given as:
\[{{100}^{\circ }}\], \[{{104}^{\circ }}\],\[{{108}^{\circ }}\], \[{{112}^{\circ }}\]……till n terms.
Now, we know that for convex polygons the sum of all interior angles as \[(n-2){{180}^{\circ }}\], where n is the number of sides.
So, we will find the sum of all the angles written above. So for that we will use the formula that sum of n terms of the AP is given by the formula \[{{S}_{n}}=\dfrac{n}{2}\left[ 2a+(n-1)d \right]\]. Here, n is the number of terms, a is the first term of the A.P. and d is the common difference.
So for the A.P. \[{{100}^{\circ }}\], \[{{104}^{\circ }}\],\[{{108}^{\circ }}\], \[{{112}^{\circ }}\]……till n terms, we have first term as \[a={{100}^{\circ }}\], common difference \[d={{4}^{\circ }}\] and there are n terms. Now, as we have sum of all interior angles as\[{{S}_{n}}=(n-2){{180}^{\circ }}\].
\[\begin{align}
  & \Rightarrow {{S}_{n}}=\dfrac{n}{2}\left[ 2a+(n-1)d \right] \\
 & \Rightarrow (n-2){{180}^{\circ }}=\dfrac{n}{2}\left[ 2\times {{100}^{\circ }}+(n-1){{4}^{\circ }} \right] \\
 & \Rightarrow (n-2){{180}^{\circ }}={{100}^{\circ }}n+{{2}^{\circ }}n(n-1) \\
 & \Rightarrow \text{18}{{\text{0}}^{\circ }}n-\text{36}{{\text{0}}^{\circ }}={{2}^{\circ }}{{n}^{2}}+{{98}^{\circ }}n \\
 & \Rightarrow {{2}^{\circ }}{{n}^{2}}-{{82}^{\circ }}n+{{360}^{\circ }}=0 \\
\end{align}\]
Now, removing the degree sign and solving the quadratic equation, as follows:
\[\Rightarrow 2{{n}^{2}}-82n+360=0\]
Now, we know that the quadratic equation of the form \[a{{x}^{2}}+bx+c=0\], will have the solutions as \[x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\]
So, now the quadratic equation \[2{{n}^{2}}-82n+360=0\] has \[a=2,b=-82,c=360\].
So then the solution will be:
\[\begin{align}
  & \Rightarrow n=\dfrac{-\left( -82 \right)\pm \sqrt{{{\left( -82 \right)}^{2}}-4\cdot \;2\cdot \;360}}{2\cdot \;2} \\
 & \Rightarrow n=\dfrac{82\pm \sqrt{3844}}{4} \\
 & \Rightarrow n=\dfrac{82\pm 62}{4} \\
 & \Rightarrow n=36,n=5 \\
\end{align}\]
So here we got \[n=5\] or \[n=36\], but here we will ignore \[n=36\], because when \[n=36\] then the 36th angle will be \[{{100}^{\circ }}+(36-1){{4}^{\circ }}={{240}^{\circ }}\] which is greater than \[{{180}^{\circ }}\] that is not possible for the convex polygon. So there are only 5 sides in the given polygon.
So the final correct answer is option (A).

Note: The sum of interior angle of the convex polygon is not \[{{360}^{\circ }}\] but is \[(n-2){{180}^{\circ }}\]. Now, this has to be kept in mind so as to make no error in calculation.