Question

a polygon have 27 diagonals how many sides does in have​

Answer 1

Answer:

Step-by-step explanation:

This is a great question! Let’s try and work it out.

Let’s assume the polygon has  n  sides. We’ll try to work out what  n  is from the information given.

First, let’s consider just 1 vertex of the polygon. How many diagonals are there in that point? Well, a diagonal is a line connecting non-adjacent vertices. So, the point itself and its 2 neighbors do not have a diagonal arriving in this point, but all other  (n−3)  vertices do. So, we have  (n−3)  diagonals arriving in this point.

Now, we have  n  such points. We might be tempted to say that  n(n−3)  is the number of diagonals, but that would be wrong! Note that every diagonal arrives in 2 vertices, one for each end-point. So, in  n(n−3) , we counted all diagonals twice.

The number of diagonals is  n(n−3)2 . Given that this is equal to  27 , we have that  n(n−3)=2⋅27=54 , or  n2−3n−54=0 . Now you could just try values of  n  until you find one that works (shouldn’t take too long) or you could solve the quadratic equation if you know how to do that. The solutions are:  n=9  or  n=−6 . Of course, the latter is not a valid solution, so we have  n=9 .

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BY Tolety Roshan  

Answer 2

Step-by-step explanation:

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