(a) A polygon with n sides has a total of 1/p*n(n-q) diagonals, where p and q are integers.

(i) find thhe value of p and q
(ii) find the total number of diagonals in a polygon with 100 sides
(iii) find the number of sides of a polygon which has a total of 170 diagonals
(b) a polygon with n+1 sides has 30 more diagonals than a polygon with n sides. find n

(i) I'm still working on this

(ii) n=100 => [n(n-3)]/2 = [100(100-3)]/2 = 4850
(iii) [n(n-3)]/2 =170 => n2-3n-340=0 <=> n = 20 and n=-17. But sides couldn't be negative => n=20 (sides)
b) Call the total number of diagonals of n+1 sides polygon is x
=> [n(n-3)]/2=x-30 and [(n+1)(n+1-3)]/2=x
<=> n=31

I) p= 2

q= 3

Find the total number of diagonals in a polygon with 100 sides

Formula to find the number of diagonals

D= n(n-3) / 2. Where n is the number of sides

D = 100(100-3)/2
D= 100 x 97 / 2
D = 9700/2
D= 4850

(a) Sure, let's get our funny math brains on!

(i) To find the values of p and q, we need to remember that there are some formulas for diagonals in a polygon. The formula for calculating the number of diagonals is given by:
1/p * n * (n - q)

To solve for p and q, we need more information. Can you provide any additional details?

(ii) Well, without the values for p and q, it's pretty hard to determine the exact number of diagonals in a polygon with 100 sides. But don't worry, I won't leave you hanging! Let's just imagine that p is as passionate as a penguin about counting diagonals and q is as quiet as a quokka. In that case, let's plug in the values (and remember that "1" times anything is just that thing):

1/p * 100 * (100 - q)

Now, since p is a cool little penguin, let's say p = 10. And since q is a shy little quokka, let's say q = 5. Let's put it all together:

1/10 * 100 * (100 - 5) = 100 * 95 = ??? Okay, I'm kidding! The answer is 9,500 diagonals. Ta-da!

(iii) Now, let's use our clown math skills to solve this one. We're given that the polygon has a total of 170 diagonals. Using our handy-dandy formula again:

1/p * n * (n - q) = 170

Without the values for p and q, it's quite the puzzle. But don't you worry, my friend! Let's use our imagination again. I'm thinking p is a prime number that loves pizza, so p = 3. And q, oh q, let's pretend q is as quiet as a mouse and set q = 4. Let's put it all together:

1/3 * n * (n - 4) = 170

Now, let's do some magical math! Multiply both sides by 3:

n * (n - 4) = 510

Solving this equation may take a bit more time, but it's totally doable! And hey, if we don't find an exact solution, we can always ask the penguin and the quokka for some pizza and have a good laugh together!

(b) Oh, now here's a fun one! We have two polygons, one with (n + 1) sides and one with n sides. The polygon with (n + 1) sides has 30 more diagonals than the one with n sides.

Hmm, let's put on our clown thinking caps and figure this out. If we call the number of diagonals in the first polygon D1 and the number of diagonals in the second polygon D2, we have:

D1 = D2 + 30

Now, using our hilarious formula from part (a):

1/p * (n + 1) * (n + 1 - q) = D1
1/p * n * (n - q) = D2

Well, that's not exactly enough information to solve it, but let's imagine p is as playful as a panda and q is as quirky as a quill. In that case, let's say p = 2 and q = 3. Now let's plug it into our equations:

1/2 * (n + 1) * (n + 1 - 3) = D1
1/2 * n * (n - 3) = D2

Now, let's substitute D1 and D2:

D2 + 30 = D2

Oops! It appears we've encountered a clown paradox. If D2 equals D2 + 30, we might need to take another approach.

But hey, let's look at the bright side — we had some fun with clown math, didn't we? Sometimes the journey is more important than the destination!

To find the values of p and q in the given formula, we can start by analyzing the pattern of diagonals in polygons and derive a general formula for the diagonals. Let's go step by step:

(a)
(i) The formula given is 1/p * n * (n - q). To determine the values of p and q, we need additional information.
(ii) To find the total number of diagonals in a polygon with 100 sides, we will substitute n = 100 into the formula and solve for the number of diagonals:

1/p * 100 * (100 - q) = ???

Since we do not have the values of p and q, we cannot determine the exact number of diagonals without further information. However, we can give you the formula to work with.

(iii) To find the number of sides of a polygon that has a total of 170 diagonals, we will rearrange the formula and substitute the given number of diagonals:

170 = 1/p * n * (n - q)

Again, without values for p and q, we cannot determine the number of sides directly.

(b) To find the value of n in a polygon with n+1 sides having 30 more diagonals than a polygon with n sides, we can set up an equation:

1/p * (n+1) * (n+1 - q) = 1/p * n * (n - q) + 30

We need the values of p and q to solve this equation and determine the value of n.

100 sides polygon have 50 diagonals