# Homework Help: No. of diagonals of a polygon

1. Jan 5, 2016

### Natasha1

1. The problem statement, all variables and given/known data
A polygon with n sides has a total of 1/p . n . (n-q) diagonals, where p and q are integers.

(i) Find the values of p and q.

2. Relevant equations

3. The attempt at a solution
Can someone just help me to start on this? I know that q = 3 and p = 2 but how?

2. Jan 5, 2016

### Samy_A

Take a polygon with n=10 (for example). How many diagonals can you draw from one of the corners?

3. Jan 5, 2016

9

4. Jan 5, 2016

### Samy_A

No. A diagonal links the corner to one of the other non-adjacent corners. How many of these are there (still with n=10)?

5. Jan 5, 2016

8

6. Jan 5, 2016

### Samy_A

No. A corner has 2 adjacent corners, and there is no diagonal linking a corner with itself. So, still with n=10, how many diagonals are there starting in one corner?

7. Jan 5, 2016

### Natasha1

If you take a triangle with 3 sides (obviously), there are no diagonals, right?

8. Jan 5, 2016

### Samy_A

Yes, that is correct. A corner of a triangle has two adjacent corners, and that's it. No possibility to draw a diagonal there.

9. Jan 5, 2016

### Natasha1

So a 4 sided polygon as only 1 diagonal, am I correct?

10. Jan 5, 2016

### Staff: Mentor

In future posts, you need to show more of an effort than this.

11. Jan 5, 2016

### Samy_A

No. A 4 sided polygon has 2 diagonals.

12. Jan 5, 2016

### Natasha1

But apart from trial and error is there any other way I can do this?

4 sides polygon
1/2 . 4 . (4-3) = 2 diagonals
5 sides polygon
1/2 . 5 . (5-3) = 5 diagonals
6 sides polygon
1/2 . 6 . (6-3) = 9 diagonals

And so on...

13. Jan 5, 2016

### Natasha1

Apologies

14. Jan 5, 2016

### Samy_A

Yes, there is. I was trying to get you there by reasoning.

So, still with n=10, how many diagonals are there starting in one corner?

15. Jan 5, 2016

### Natasha1

4 sided polygon
1/2 . 4 . (4-3) = 2 diagonals
5 sided polygon
1/2 . 5 . (5-3) = 5 diagonals
6 sided polygon
1/2 . 6 . (6-3) = 9 diagonals
...
10 sided polygon
1/2 . 10 . (10-3) = 35 diagonals

16. Jan 5, 2016

### Natasha1

Is there anyway I can do this problem without doing it like i have, guessing what p and q are...

How can one prove it?

17. Jan 5, 2016

### Samy_A

By reasoning how many diagonals there are. It is possible, it is easy, but you have to try it.

Outline:
Take one corner, and compute the number of diagonals it lies on. We were almost there: there is a diagonal linking that corner to any other corner, except itself and its two adjacent corners. That makes n - 1 - 2 = n -3 diagonals from that one corner.
Now, this is the case for each one of the n corners. So now you can compute the total number of diagonals, but don't forget that each diagonal connects two corners.

Good night.

18. Jan 5, 2016

### Natasha1

Got it thanks!