Question

If a polygon of n sides has 275 diagonals, then n is equals to
A) 25
B) 35
C) 20
D) 15

Answer 1

Hint: Diagonals are the segments drawn from one vertex to the exactly opposite vertex. We know that as the number of sides of a polygon increases the number of diagonals we can draw also increases. We have a formula in that case with the help of which we can exactly determine the number of diagonals a polygon can have if the number of sides are given to us and even the number of sides if the number of diagonals are given. We will use the formula here.

Formula used:
If n is the number of sides of a polygon then the number of diagonals is given by,
\[\dfrac{{n\left( {n - 3} \right)}}{2}\]

Complete step by step answer:
We are given the number of diagonals of a polygon.
As we know the formula above given the number of diagonals a polygon has is based on the number of sides but we have to find the number of sides.
So we will equate it with the number of diagonals given to us.
\[\dfrac{{n\left( {n - 3} \right)}}{2} = 275\]
Now cross multiply the 2,
\[n\left( {n - 3} \right) = 275 \times 2\]
On multiplying we get,
\[n\left( {n - 3} \right) = 550\]
Now we have to find two numbers such that one is n and other is 3 less than n and their product will be 550.
So such a pair is 25 and 22 such that their product is 550.
Thus the value of n is 25.
The diagonals of the polygon are 25. Thus option A is correct.

Note:
Note that, in the formula n is the number of sides and not the number of diagonals whereas the whole formula after calculation gives the number of diagonals.
In order to find such two numbers we will use the options only. Choose values, find three less than the number and multiply. Then check if it equals 550 or not. Like the trial-and-error method.
If not then we can multiply the bracket and will get a quadratic equation and on solving which we will get the value of $n $. But it is too tedious and time consuming in this case, so it's better to go the way above.