The number of diagonals D I a polygon of n sides is given in the formula D=n(n-3)/2.
Use the formula to find:
A. The number of diagonals in a polygon with 9 sides.
B. The number of sides of a polygon with 90 diafonals.
just plug in the numbers. Should not be too hard.
A. To find the number of diagonals in a polygon with 9 sides, we can use the formula D = n(n-3)/2.
Substituting n = 9 into the formula, we get:
D = 9(9-3)/2
D = 9(6)/2
D = 54/2
D = 27
Therefore, a polygon with 9 sides has 27 diagonals.
B. To find the number of sides of a polygon with 90 diagonals, we can rearrange the formula D = n(n-3)/2 to solve for n:
D = n(n-3)/2
90 = n(n-3)/2
Multiplying both sides of the equation by 2, we get:
180 = n(n-3)
Expanding the equation, we get:
180 = n^2 - 3n
Rearranging the equation, we get:
n^2 - 3n - 180 = 0
Now, we can solve this quadratic equation to find the value(s) of n. Factoring the equation or using the quadratic formula, we find that:
(n - 15)(n + 12) = 0
Therefore, n = 15 or n = -12. However, since the number of sides of a polygon cannot be negative, the only valid solution is n = 15.
Therefore, a polygon with 90 diagonals has 15 sides.
To find the number of diagonals in a polygon given the number of sides, we can use the formula D = n(n-3)/2, where D represents the number of diagonals and n represents the number of sides.
A. The number of diagonals in a polygon with 9 sides:
By substituting the value of n as 9 into the formula, we have:
D = 9(9-3)/2
D = 9(6)/2
D = 54/2
D = 27
Therefore, a polygon with 9 sides has 27 diagonals.
B. To find the number of sides of a polygon with 90 diagonals, we need to rearrange the formula to solve for n:
D = n(n-3)/2
First, multiply both sides of the equation by 2 to get rid of the denominator:
2D = n(n-3)
Next, distribute n to get a quadratic equation in the form of:
2D = n^2 - 3n
Rearrange the equation by bringing all terms to one side:
n^2 - 3n - 2D = 0
This is now a quadratic equation that can be solved using either factoring, completing the square, or the quadratic formula. Since we are looking for the number of sides, which must be positive, we can discard any negative solutions.
After obtaining the solutions for n, we can check which one makes sense in the context of the problem. Since the number of sides of a polygon must be an integer, we should round our answer to the nearest whole number.
Finally, by substituting the value of D as 90 into the formula, we can solve for n:
n^2 - 3n - 2(90) = 0
Simplifying the equation, we have:
n^2 - 3n - 180 = 0
Using factoring or the quadratic formula, we find that the two possible solutions for n are approximately -12.937 and 15.937. However, since the number of sides must be positive, we can round the solution to the nearest whole number.
Therefore, a polygon with 90 diagonals has approximately 16 sides.