SOLUTION: polygon k has 36 more diagonals and 3 more sides than polygon L. How many sides do each of these polygons have

Algebra ->  Polygons -> SOLUTION: polygon k has 36 more diagonals and 3 more sides than polygon L. How many sides do each of these polygons have      Log On


   

Question 1003640: polygon k has 36 more diagonals and 3 more sides than polygon L. How many sides do each of these polygons have
Found 2 solutions by ikleyn, KMST:
Answer by ikleyn(52800) About Me  (Show Source):
You can put this solution on YOUR website!
.
Polygon K has 36 more diagonals and 3 more sides than polygon L.
How many sides do each of these polygons have?
----------------------------------------------------------------

An  n-sided convex polygon has  %28n%2A%28n-3%29%29%2F2  diagonals.

Let  n  be the number of sides of the polygon  K.
Then it has  %28n%2A%28n-3%29%29%2F2  diagonals.

The polygon  L  has  n+3  sides and  %28%28n%2B3%29%2An%29%2F2  diagonals.

We are given that

%28%28n%2B3%29%2An%29%2F2 - %28n%2A%28n-3%29%29%2F2 = 36.

Solve this equation for  n.

(n+3)*n - n*(n-3) = 72,

6n = 72,

n = 72%2F6 = 12.

Answer.  K  is  12-sided polygon and  L  is  15-sided  polygon.


Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
k= number of sides of the polygon K.
So, k%2B3= number of sides of polygon L.

A polygon with n sides has n vertices.
Each vertex can be connected to each of the other n-1 vertices by a segment.
Two of those segments will be sides, connecting adjacent vertices.
The other n-1-2=n-3 segments will be diagonals.
Each of the n vertices is involved in n-3 diagonals,
but since each diagonal involves 2 vertices,
the total number of diagonals for polygon with n sides is
n%28n-3%29%2F2 .

The total number of diagonals for polygon with k sides is
k%28k-3%29%2F2 .
The total number of diagonals for polygon with k%2B3 sides is
%28k%2B3%29k%2F2 .
The problem says that
%28k%2B3%29k%2F2=36%2Bk%28k-3%29%2F2 .
We solve for k .
%28k%2B3%29k%2F2=36%2Bk%28k-3%29%2F2
%28k%2B3%29k=72%2Bk%28k-3%29
k%5E2%2B3k=72%2Bk%5E2-3k%29
3k=72-3k%29
6k=72%29
k=72%2F6%29
k=12%29
k%2B3=15
Polygon K has highlight%2812%29 sides, and polygon L has highlight%2815%29 sides.