Question

the equation d=1/2n (n-3) gives the number of diagonals D for polygon with n sides. use this equation to find the number of sides n for a polygon that has 65 diagonals

Answer 1

Answer: 13

Explanation:

Given : The equation
`d=(1)/(2)n(n-3)` gives the number of diagonals d for polygon with n sides.

To find the number of sides n for a polygon that has 65 diagonals, we substitute the value of d= 65 in the given equation, we get

`65=(1)/(2)n(n-3)`

Multiply 2 on both sides , we get

`n(n-3)=130\\\\\Rightarrow\ n^2-3n-130=0\\\\\Rightarrow\ n^2-13n+10n-130=0\\\\\Rightarrow\ n(n-13)+10(n-13)=0\\\\\Rightarrow\ (n-13)(n+10)=0\\\\\Rightarrow\ n= -10\ or\ n= 13`

But number of sides cannot be negative, so the number of sides n for a polygon that has 65 diagonals = 13

Answer 2

To solve this problem you must apply the proccedure shown below:

1. You have that:

- The equation d=1/2(n(n-3)) gives the number of diagonals for the polygon.

- The polygon that has 65 diagonals..

2. When you clear n, you obtain:

d=n(n-3)/2
d=(n^2-3n)/2
2x65=n^2-3n
n^2-3n-130=0

3. When you solve the quadratic equation, you obtain:

n=13

Therefore, the answer is: 13 sides.